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MATHEMATICAL TABLES: 

CONTAINING 

COMMON, HYPERBOLIC, and LOGISTIC 

LOGARITHMS. 

ALSO 

SINES, TANGENTS, SECANTS, and VERSED-SINES, 

BOTH 

' natural/and logarithmic. 

TOGETHER WITH 

several other tables useful in mathematical 

calculations. 

To which is prefixed, 

A large and original history 

O P T H E 

DISCCrVERIES AND WRITINGS RELATING 

TO THOSE subjects/ 

\V I T H T H E y 

COMPLEAT description and USE.^of the TABLES.^ 



By CHARLES HUTTON, LLD. F.R.S. 1 1 '- ^ 

AND 

Profeffor of Mathematics in the Royal Military Academy, Woohvlch. 



LONDON: 

PRINTED roR G. G. J. and J. ROBINSON, and R. BALDWIN, 

PATE R N O S T E R - R O \\\ 

M DCC LXXXV. 

(< \,' 






« ^'sim^ 



T O T H E 



si 



REV, NEVIL MASKELYNE, D.D. F.R.S. 

ASTRONOMER R O Y A L, &c. 

i 

> IN 

TESTIMONY OF RESPECT, 

F O.R HIS 

EMINENT LEARNING AND ABILITIES, 

• AND FOR HIS 

XAUDABLE AND DISINTERESTED PROMOTION 

AND ENCOURAGEMENT OF THESE, AS WELL AS 
ALL OTHER USEFUL SCIENCES ; 

AND OF GRATITUDE, FOR HIS 

GENEROUS. ADVICE 

AND ASSISTANCE IN THIS WORK, 
AND ON VARIOUS OTHER OCCASIONS; 



THESE SHEETS ARE HUMBLY INSCRIBED 



BY HIS MOST OBEDIENT SERVANT, 



The AUTHOR 






( » > 



PREFACE. 

THE very ample introduction^ prefixed to the 
following colleAion of Mathematical Tables, 
has fuperccded the neceffity of ufing many words here 
by way of preface, and has left me little more to 
mention thkn the neceffity and occafion of this work, 
"with fome account of the contents and mode of execu- 
tion. 

The undertaking was occafioned by the extreme in- 
correftnefs of the 5th, or laft, edition of Sher win's 
once very ufeful book of tables. Finding, as well 
from the report of others, as from my own experience, 
that that edition (to fay notliing of the very improper 
;ilteration in the form of the table of fines, tangents^ 
and fecants) was fo very incorredlly printed, the errors 
being multiplied beyond all tolerable bounds, and 
no dependence to be placed on it for any thing of real 
pradlice, I was led to undertake the painful office of 
preparing a correA edition of another fimilar work. 
And I was lucky enough to meet with a bookfcller of 
fufficient fpirit to be at the great expence of printing 
the book» as well as to allow me what I demanded for 
my trouble in preparing it ; which demand however, 
was nothing adequate to the great labour attending it, 

A as 



VI 



PREFACE. 



as I was well aware that the profits of the book would 
not afford him the means of rewarding my pains. 

I have in the firft place, therefore, ufcd all the means 
in my power to render the work corredt. I began by 
collating the 3d, or beft edition of Sherwin's tables, 
with feme others of the moft pcrfe<fl works of the fame 
kind, as Briggs*s, Vlacq's, Gardiner's quarto book, 
&c : by which* means I deteftcd many errors in each of 
them, which had not before.been difcovered ; and of 
thefe between twenty and thirty were in the two edi- 
tions of Gardiner's work, printed at London in' 1742, 
and at Avignon in 1770 -, the errata of which two books 
arc here printed at the end of our tables. But, befidcs 
detedling many unknown,errors in my copy of the faid 
3i Edition of Sherwin, which was no more than what 
I expeftcd, I difcovered, with no fmall furprize, that 
the laft figures in the table of logarithms were not uni- 
formly true to the neareft unit, except in a very few 
pages at the beginning and end of the table; al- 
though Mr. Gardiner, the editor of ^that edition, had 
made the table correft in that refpedt in his own quarto 
work before- mentioned, which was alfo printed in the 
fame year 1742 with the faid third edition of Sherwin! 
The errors from this caufe amounted to feveral thou- 
fands ; and they have continued to run through all the 
editions of Sherwin evef fince that time ! But I have here 
corrected them. Nor have I employed lefs attention in 
correcting the prcfs, than in previoufly correcting 
the copy; ^ery proof having been feveral times read 
over, and compared with the beft of the books before- 
mentioned. 



PREFACE. vli 

But in giving this edition to the world> I was not 
fatisfied with barely making it correcl, I was aware 
that the materials themfelves might be much improved ; 
and I have accordingly enlarged, or otherwife greatly 
amended them, in various refpcdls. Among the im- 
provements of the old materials, may be reckoned the 
following : namely, in the large table of logarithms, 
the proportional parts, near the beginning, are more 
conveniently arranged, being now all placed in the 
fame opening of the book where their correfpond- 
ing differences occur : The logarithms to fixty one ^ 
figures are brought to their proper place in the book, and 
more conveniently difpofed all in one page : The large 
table of fines, tangents and fecants, is more commo- 
dioufly arranged, and rendered more diftinft and con- 
venient for ufe ; the natural fines, tangents, fecants and 
verfcd fines, being all feparated from the others, and 
placed all together on the left-hand pages, and the 
logarithmic ones facing them on the right-hand pages ; 
the common differences, in both, fet between the 
two columns to which each of them anfwers; and the 
verfed fines here introduced into their proper place 
in the fame pages with the fines, tangents and fecants. 
Befides thcfe, there are fome other alterations in the 
new tablts here given, and the reader will find a num- 
ber of very important improvements in the defcription 
and ufe of the whole; efpecially in the arithmetic of 
logarithms, and in the refolution of plane and fpherical 
triangles, according to the prefent improved methods 

A 2 of 



viii PREFACE. 

of calculation ufed by the Aftronomcr Royal, and other 
perfons the mofl experienced in thefe matters. 

The improvements in the tables, by the introduftion 
of new matter, are both great and numerous. The tables 
numbered 2, 3, and 4 are here added, being an entire 
new fet, with their differences, for finding numbers and 
loj^arithms to twenty places. The columns of comr 
nion differences, in the pages of natural fines &c, are 
now firfl introduced : As are alfo the tables^ of hyr 
perbolic and logiflic logarithms ; the logarithmic fines 
and tangents for every fecond, in the firfl two degrees of 
the quadrant ; together with a table of the lengths of 
arcs^ a table to change common and hyperbolip loga- 
rithms from the one to the other; &c : the ufes an4 
exemplifications of the yrhole being very amply de? 
tailed. 

But the greatefl alteration of all, is the very exten? 
five and new introduAion here given, inflead of the 
former inadequate and heterogeneous one, confifling 
pf about 180 pages of new matter, on a methodical 
plan, containing the hiflorical accoqnt and defcription 
of all trigonometrical writings, and the tables relating 
to that fubjedl, both natural and logarithmic 5 befides 
the compleat ufe of our own tables. Inventions are 
here afcribed to the proper authors, and their methods 
and improvements d'efcribed and compared. This 
hiflorical defcription will evidently appear to be the 
refult of immenfe labour and reading. And indeed I 

Jjavc 



PREFACE. 

have painfully gone over all the books which are here to 
minutely defcribed ; and that defcription with a detail 
in fome degree adequate to their great merits ; eipe« 
cially the works of Napier, Briggs, Kepler, &c; which 
was the more neceifary, as the writings and methods 
of thofe great mafters ha4 not been any whej-e properly 
defcribed and difcriminated, although they are in them- 
felves highly curious and important; 

Thefe readings and commentaries have been carrie4 
on to an extent far beyond what was at iirft intendeds 
But the tables having been in the prefs for the fpace of 
feven or eight years, I had thereby an opportunity of 
coUeifling and examining a ftill greater number of 
books ; fo that I was gradually led on, and my view^ 
and plans rendered ftill more extenfive and compleat. 
This delay, therefore, though in niany refpedls it 
proved very inconvenient and difagreeable, has at 
length given the occaf^on of rendering thefe cqmmen-p 
taries more perfe<9; and fatisfadlory. 

Befides what immediately relates to trigonometrical 
fubjeifl^j the reader will here find many other curious 
and uncommon articles^ relating to the feveral authors 
and their difcoveries, which have occurred in the 
courfe of my reading, and which appeared of too much 
confequence to be paffed over unnoticed, in the analyfis 
qf their feveral compofitions. Among thefe is thedifco-' 
very onhe firft author of the binomial theorem, and the 
^ffercntial^niethod, which are due to Mr. Henry Briggs, 

whofe 



af PREFACE. 

whofe writings are replete with ingenious and original 
matter^ and are well dcferving to be more generally 
known and fludied than they have been for fome time 
paft. 

This long courfe of examination and defcription, 
however, having been carried on for fo many years, at 
different intervals, and interrupted by various avoca- 
tions, and by bufinefs of different kinds, it will be no 
wonder if this circumftance may have occafioned fome 
inequalities in the ftile and compoiition of this hiflory i 
and for which therefore, ihould any fuch appear, it is 
hoped the occafion will plead an apology. 



P. S. Since my Hiftory of Trigonometrical tables, in the foUowing in- 
trodudtiooy was printed, there has been publifhed, in the Philofophical 
Tranfadions for the year 1784, a paper of mine concerning a projedt ibr 
the trigonometrical tables to be confiruded oh a new plan, namely, in 
which the arc of the quadrant is divided into aliquot parts of the radius, of 
according to the real lengths of the arcs. Which conftru^tion is now in 
fome degree of fbrwardneu, as myfelf and feveral afliQants have been 
dofely engaged in the execution or it e?er fince. 



Afhort 



( XI ) 



A (liort abflra£l of the principal contents^ may be as follows : 



!• In the IntroduBion* 



HiHory of trigonometrical Cables 
before the invention of loga- 
rithmsy with the various me- 
thods of condrudtion . I 
On the word ^nus ■■■ ■ 17 
Hiftory of logarithms ■ 20 
Nature of logaiithms — 21 
Invention of logarithms — 23 
Different forts of logarithms 24 
Con{lru6Uon of logarithms — • 41 
By Napier '«*— — 41 
Kepler — — — 48 
Briggs — ^ ■ 60 
Brigg8*s Trigonomctria Britan. 73 
Relation between logarithms and 

certain curves — p 81 

Gregory's conftruftion — 84 

Mercator*8 Logarirhmo-technia 84 

Gregory's Excrcit. Geometricae 94 

Sit I. Newton's methods — 99 



Page 

Halley'a ■ ■ » 103 
Sharp's — — ■ 107 
Cotes's Logometria — ^ 109 
Taylor's conftrudion -^ iiz 
Long's method ■ ■ 114 
Jones's — ■ 116 
Reid&c — — X18 
Dodfon's Anti-log. canon 119 
Defcription and ufe of logarith- 
mic tables 122 
Definition and notation 122 
Properties of logarithms — • 124 
Confhrudlion of logarithms 1 24 
Defcription and ufe of our tables x 26 
Of our large table —— 126 
Logarithmical arithmetic >x3i 
Of the table to 20 places j 34 
Of the table to 61 places ^ 1 39 
Of the hyperbolic logarithms 143 
Of the logifUc logarithms 144 



2. In the TahifS them/elves^ 



Tab. 

1. Logarithms from t to 1 00000 x 

2. Logarithms &c to 20 places 187 

3. Id. with differences 198 
4« Numbers to logarithms to 20 

places 201 

5. Logarithms to 61 places 204 

6. Idt with differences -^ 208 

7. Hyperbolic logarithms 209 

8. Logiilic logarithms — 213 

9. Sines and tangents to fecouds 2 1 8 



Tab. 

10. Natural and logarithmic 

lines, tangents, fecants^ and 
▼erfed fines 

1 1. Travcrfe tabic •— 
X2, Lengths of arcs 

1 3. Table to change common 

and hyperbolic logarithms 
from one to the other 

14. Points of the compafs 

15. Errata in Gardinei's loga- 

rithms 



248 
338 
340 



341 
341 

34* 



< «ii ) 



ERRATA IN TBE LOGARITHMS. 



Page 


Numb. 


Log. 


34 - 


23662 


0J15 


34 -. 


2 37i54 - 


7368 


38 - 


45519 - 


8637 


38 - 


25808 - 


7S44 


44 - 


28565 - 


8342 



58 In the pro. parti, under the 

dif. I2Z, for 63 read 6f.' 
119 lb. under 65, for 53 read js. 



• '<~T 



INTRODUCTION. 



»■ ifc 



1 OF TRiGONaMETAICAL TABLESj &c. 



NECESSITY, the fruitful mother of moft ufcfu] inventions, gave 
birth to the various numerical tables which compofe the following 
work. Aftronomy has been cultivated from the earlieft ages* The 
progrefs of that fciehce, requiring numerous arithmetical computa- 
tions of the fides arid angles of triangles, both plane and fpherical, 
gave rife to trigonometry ) for thofe frequent calculations fuggeftcd the 
iiecei&ty of performing them by the property of finiilar triangles } and 
for the ready application of this property, it was neceJTary that certain 
lines defcribed in and about circles, to a determinate radius^ fhould be 
computed, and difpofed in tables. Navigation^ and the continually 
improving accuracy of ai^ronomy, have alfo occafioned as perpetual an 
increafe in the accuracy and extent of thofe tables. And this it is 
Evident muft ever be the cafe, the improvement oJF trigonometry uni« 
formly following the improvement of thofe other ufeful fciences, for 
the fake of which it is moire efpecially cultivated. 

't'he ancients performed their trigonometry by means of the chords 
of arcs, which with the chords of their fupplemental arcs, and the con« 
fiant diameter, formed all fpecles of right-angled triangles. Beginning 
- with the radius, and the arc whofe chord is equal to the radius, they 
divided them both into 60 equal parts, and eftimated all other arc3 and 
chords by thofe parts, namely all arcs by 6oths ofthat arc> and ali 
chords by 6oths of its chord or the radius : At lead this method is as 
i!>Id as the writings jof Ptolemy, who ufed the fexagenary arithmetic for 
this diviiion of chords and arcs^ and for aftronomipal purpofes* — And 
this by-the-byeihews the reafon why the whole circumference is di« 
Vided into 360, or 6 times 60, equal parts or degrees, the whole circum* 
ference being equal to 6 times the nrft arc whofe chord Is equal to the 
tadius : Unlefs perhaps we are to feck for the divifion of the circle in 
the number of days in the year; for thus, the ancient year confifting of 
360 days, the fun or earth in each day defcribed the 360th part of th^ 
orbit; and thence might arife the method of dividing every circle into 
360 parts \ and, radius i>eing equal to the chord oImSo of thofe parts^ 
the fexageflmal divifion both of the radius and of the parts might thence 

B arife. 



a ftlSTOfeYOF 

arife. Trigonometry however muft have been cultivated long tie(o{V 
the time of Ptolemy; and indeed Theon, in his commentary on 
Ptolemy's Almagefl, 1. i. ch. 9, mentions a work of the philofopher 
Hipparchus, written about a century and a half before Chrift, confift- 
ing of 12 books on the chords of circular arcs ; which muft have been 
a tce^tife on trigonomotry. Add Menelaus alio, in the firft oentury 
Ct Chrift, wrqte 6 books concerning fubtenCes or chords of arcs. Hef 
ufed the word nadir (of an arc), which he defined to be the right line 
fubtending the double of the arc; fo that his nadir of an arc, was the 
double of our fine of the fame arc ; and therefore whatever he proves 
of the former, may be applied to the latter, fitbAituting the double fine 
for the nadir. 

The radius has finc^ been decimally divided ; but the fexagefimal 
divifions of the ^rc have cpntinued in ufe to this day* Indeed our 
countrymen Briggsand Gellibrand, having a general diilike to a]} (exa^ 
gefimal divifions, made an attempt at fome reformation of this cuftom, 
by dividing the degrees of the arcs, in their tables, into centefms or 
hundredth parts, infiead of minutes or 60th parts. The fame was alip 
lecommende^ by Vieta and others ; and a decimal dtvifion of the whole 
quadrant might perhaps fbon have followed, had it not be^n for the 
tables of Vlacq, which canie out a little after, to every 10 feconds, 
or 6th part of a minute.-^But the compleat reformation, would be, to 
exprefs all arcs by their real lengths, i^amely in equal parts of the ra- 
dius decimally divided : of which more in its proper place. 

It is not to be doubted that many of the ancients wrote on the fub- 
jed of trigonometry, as being a nece({ary part of aftronomy ; although 
few of their labours on that branch have come to our knowledge, and 
fiill fewer of the writings themfelves have been handed down to us. 

We ace in pofleffion of the 3 books of Menelaus on fpherical tri^o-» 
jiometry ; but the 6 books are loft which he wrote on chords, being 
probably n treatife on the conftru£lion of trigonometrical tables. 

The trigonometry of Menelaus was much improved by Ptolemy 

{Claudius rtolemaeusj the celebrated philofopher and mathematician. 
le was born at Peluiium, taug;ht aftronomy at Alexandria in Egypt^ 
and died in the year of Chrift 147, being the 78th year of his' age. In 
the firft book of his ^Imageft, Ptolemy delivers a table of arcs and 
chords, with' the method of conftruAion. Tkis table contains 
2 columns : in the ift are the arcs to every half degree or 30 minutes | 
in the 2d are their chords, exprefled in degrees, minutes and fec^onds, 
cf which degrees the radius contains 60 ; and in the ^d column are 
the dtfterences of the chords anfwering to i minute of the arc9, or the 
30th part of the difierences between the chords in the 2d column. In 
the conftruftion of this table, among others, Ptolemy fliews, ipr the 
iirft time that we know of, this property of any quadrilateral inicribed 
in a circle, namely that the refiangle under the two diagonals, is equal 
to the fum of the two redangles under the oppofite fides. 

This method of computation, by the chords, continued in ufe tiH 
•bout the middle centuries after Chrift; when it was changed for that 
of the fines, which w;ere aboMt that time introduced into trigonometry 
J^J the Arsibiansy who in. pthei: rcfpe£ts much improYpd this fcience» 

whicli 



TRIGONOMETRICAL TABLES, &c. ^ 

Mkich they received from the Greeks^ introducing, amofig other thln^r^, 
the three or four theorems, or axioms, which we ufe at prefent as the 
foimdation of our modem trtgononfietry. 

The other great improvements that have been made in this branch, 
are due to the Europeans. Thefe improvements they have gradually 
introduced fince they received this fcience from the Arabians. And 
ilthou^h thefe latter pftople had h>ng uied the Indian or decimal fcafe 
of arichmettc, it does not appear that they varied from the Greek at 
Texagefimal divifioh of the radius, by which the choirds and Anes weiib 
ekprefled. 

This arlteration is faid to have been firft made by George Purblch^ 
who was fo called from his being a native of a place of that name be« 
tweeh Anftria and Bavaria. He was born in 1^23, ftudied mathe- 
matics and aftronomy at the univerfiiy of Vienna, where he was after- 
wards profeflbr of thofe fciences, though but for a (hort titne, th^ 
learned world quickly fufFering a great lofs l^y his immature dfeiitb^ 
which happened in 1462, at the age of 39 years only. Parbach, be- 
fides enriching trigonometry and aftronomy with feveral new tables, 
theorems, and obfervations, fuppofed the radius to bedivided into6ooooo 
%qual parts, and computed the fines of the arcs, for every 10 minutes, 
in fuch equal parts of the radius, by the decimal notation. 

This projed of Purbach was compleated by his difciple, compahioit, 
and fucceflbr John Muller, or Regiomontanus, who was fo called froih 
the place of his nativity^ the little tower of Mons Regius, or Konitig- 
berg in Franconia, where he vns born in the year 1436. Regioifion- 
tanus not only extended the fines to every minute, the radius being 
60000O9 as dengned by Purbach, but afterwards, difliking th^t fcheme, 
9S evidently imperfe£fr, he computed them likewife to the radius 
loooooo, ror every minute of the quadrant. He alfo introduced the 
tangents into trigonohietry, the canon of which he called feecundus 
becaufe of the many and greiat advantages arifin^ from them. Befides 
thefe he enriched trigoncfmetry with many theorems and precepts. 
Through the benefit of all thefe improvements, except for the ufe of 
logarithmr, the trigonometry of Regiomontanus is btit lirtle inferior 
to that of our own time. His treatife, on both plane and ipheric^I 
trigonometry, is in 5 books; it was written about the year 1464, 
and printed in folio at Nurembttrg in 1533. And in the 5th book are 
vartous problems concerning redilinear triangles, fonie of which are 
tefolved by means of algebra : a proof that this fcience was not wholly 
unknown in Europe before the treatife of Lucas de Burgo. Regio- 
montanus died in 1476 at the. age of 40 years only, being then at 
Rome, whither he had been invited by the Pope, to affifl in the refor* 
mation of the calendar, arid was fufpej^ed to have been poiibned there 
|>y the fons of George Trebizonde, in reVenge for the death of their 
father, which was faid to have. been caufed by the grief he felt on ac- 
count of thfe criticifms made by RegiomontantiS on his tranflation of 
Ptolemy's Almageft, 

Soon after this, leveral other mathematiciaite contributed to the im- 
provement -of trigonohietry, by extending and enlarging the tables, 
though f«iv of their works have been primed \ a^d particuhurly John 

fi % Werner 



4 HISTORYOF 

Werner of Nuremburg, who was born in 1468 and died in 1529, ali4 
who is faid to have written five books on triangles. 

About the year 1500, Nicholas Copernicus, the famous modern r&r 
ftorer of the true folar fyftem, Wi'ote a brief treatife on trigonometry 
both plane and fpherical, with the defcription and conftru£tton of the 
canon of chords, or their halves, nearly ia the manner of Ptolemy ; to 
which is fubjoined a canon of fines, with their differences, for ev^y 
10 minutes of the quadrant, to the radius ^00000. This tra6t is in-? 
iertedjnthe firft hook of his Rev^luiioms OrbiumCceUftium^ firft printed 
in folio at Nuremburg 1543. It is remarkable that he does not call thefa 
lints Jines^ hut femiffis fkbtenfarumy namely of the double arcs, — Coper- 
nicus was born at 1 horn in 1473, and died in I543* 

In 1553^ was publiflied the Canon Foecundus^ or table of tangents, of 
Erafmus Reinhold, profeflbr of mathematics in the academy of Wur* 
temburg. He was born at Salfieldt in Upper Saxony, in the year 
2511, and died in 1553. 

To Francifcus Maurolycus, abbot of Meffina in Sjcily, we owe the 
introdu6iion of the Tabula Benefica^ or canon of fecants, which came 
out about the fame time, or a little before. But Lanlbergius ei'rone* 
oufly afcribes this to Rheticus. And the tangents and iecants are 
both afcribed to Reinhold, by Briggs, in his Mathematica ab antiquU 
minus cognita^ (pa* 30. Appendix to Ward's lives of the profefTors of 
preiham cojlege.) 

Francis Vieta was born In 1540 at Fontcnai, or Fontenai-le-Comte, 
in Lower Poitou, a province of France. He was mafter of requefls at 
' Paris, where he died in 1603, being the 63d year of his age. Among 
other branches of learning in which he excelled, he was one of the 
mod refpeftable mathematicians of the i6th century, or indeed of any 
age. His writings abound with marks of great originality, and the 
fineft genius, as well as intenfe application. Among them are feve- 
ral pieces relating.to trigonometry, which may be found in the collec- 
tion of his works publiflied at Leyden in 1646, by Francis Schooten^ 
befides another large and feparate volume in folio, publifhed in the 
author's life time at Paris in 1579, containing trigonometrical tables 
with their conftruflion and ufe ; very elegantly printed, by theAing^ 
mathematical printer, with beautiful types and rules, the differences 
of the fines, tangents and fecants, and fome other parts, being printed 
with red ink, for the better diftin<ftion j but inaccurately executed, as 
he himfelf teftifies in pa, 323 of his other works above-mentioned. 
The firft part of this curious volume is intituled Canon Mathematicus 
Jeu ad Triangula^ cum Jppendictbusy and contains a great variety of 
tables ufeful in trigonometry. The firft of thefe is what he more pe- 
culiarly calls Canon Mathematicus, feu^ ad Triangula^ which contains 
all the fines, tangents, and fecants for every minute of the quadrant, 
to the radius ioo,oco, with all their differences ; and towards the end 
of the quadrant the tangents and fecants are extended to 8 or 9 places 
of figures. They are arranged like our tables at prefent, increafing on 
the left-hand fide to 45 degrees, and then returning upwards by the 
right-hand fide to 90 degrees ; fo that each number and its complement 
Rand together on the fame Hoc,' But here the can^n of what we now call 

tangents 



TRIGONOM ETRieAL TABLES, &a, j| 

I 

tangents is denominated foecunius^ and that of the fecants facunitj^ 
fimus*, For the general idea prevailing in the form of thefe tables, is^ 
not that the lines reprefented by the numbers are thofe which are 
drawn in and about a circle, as fines, tangents and fecants, but the 
three fides of right-angled triangles^ this being the way in which thofe. 
lines had always been confidered, and which ftill continued for foiqe 
time longer. And therefore he confiders the canon as a feries of plane 
right-angled triangles, one fjde' being conftantly ibo>ooo^ or rather 
as three ftries of fuch triangles, for he makes a diilin^ feries for each 
of the three varieties, namely, according as the hypotenuGe or the b.afo 
or the perpendicular is reprefented by the conftant number 100,000, 
which isfimilar to th^ radius, Makipg e^ch fide conftantly 100,000, 
the other two (ides are computed to every magnitude of the acute angle 
at the bale, from i minute up to 90 degrees or the whole quadrant* 
Each of the three feries therefore confifts of two parts, as reprefenting 
the two variable fides of the triangle* When the hypotenufe is made the 
conftant number 100,000, the two variable fides of the triangle are the 
perpendicular ^d bafe, or our fine and cofine ; when the bafe is 100,000^ 
the perpendicular and hypotenufe are the variable parts,, forming the 
canon fmcundus li fcecundij^mus^ of oqr tangent and fecant ; and when 
the perpendicular is made the conftant 100,000, the feries contains the ' 
variable bafe and hypotenufe, or alfo canon fcecundus ^ fcecundijjimus^ 
or our cotangent and cofecant. Of courfe therefore the table confifts 
of 6 columns, 2 for each of the 3 feries^ befides the two columns oi\ 
the light and left for minutes, from o to 6q in each degree. 

The fecond of thefe tables is fimilar to the firft, but all in rational 
numbers, confifting, like 1/, of 3 feries of 2 columns each, the radius^ 
or conftant fide of the triangle, in each feries, being 100,000, as be- 
fore, and the other two fides accurately exprefTed in integers and rational 
vulgar fradlions. So that we have here the canon of accurate fines, 
tangents and feeants, or a feries of about 4^00 rauional right- aiiglcd 
triangles. But then the feveral correfponding arcs of the quadrant, 
or angles of thofe triangles, are not exprefTed. Inftead of them are 
jnferted, in the firft column next the margin, a feries of numbers de« 
ereafing from the beginning to the end of ike" quad rant, which are 
called numeri primi bafeot* It is from thefe numbers that Vieta con* 
flruds the fides of the 3 feries of right-angled triangles, one fide in each 
feries being the conftant number 100,000, as before. The theorems by 
which thefe feries of rational triangles are computed from the numeri 
frimi bafeos^ or marginal numbers, are inferred all in one page at the 
end of this 2d table, and in the modern notation they may be briefly 
exprefled thus. Let p be the primary or marginal number on any line, 
and r the conftant radius dl* number ioc,ooo; then if r denote the 
hypotenufe of the right-angled triangle, the perpendicular and bafe, 
or the fine and cofine, will be refpeftively 

— C-— and r— — --!L, (which lafl we may reduce to ^ ~^ r) ; . 
^v|ie|i r denotes the bafe pf the right-angled triangle, the perpendicular 



^ 



HISTORY OP 






$^i hjrpotetmre) vr the tangent and (ecant* M expreflcd by 

L,i_ ' { *™ r "4- IJ2^> (which laft We may reduce to 

and when r denotes the perpendicular of the right-angled triangle, th^ 
^afe and hy|)otenuiey or die cotangent andcofecant, are then expreiTed by 

l/r-f (or^ r) and |^r + ^ (or i£±Lr). 

P t , P P 

to that Victa's general values will be as we have here colleSed them 
tbgether in the Allowing exprefiions immediately under the words fine, 
conne, &c ; and juft below Vieta's forms I have here placed the others 
to which they reduce and are equivalent, which are more contracSted, 
but not lb well adapted to the expeditious computation as Vieta's forms. 



,Jm 



Sine 
pr 

p 



Cofine 



r— 






Tangent 



[If in in , I [| 



r4 



Secaoc 
ar 



i/*-« 



Cotati|eiit 



} \ 



Co^bcant 



P 



't 



u 



> .'t: 



•«* 



All thefe expreiSons it is evident are rational ; and by aiTuming p of 
different values, froin the firft theorems V ieta computed the correlpond* 
ing fides of the tri^ngles^ and fp cxprefTed them all in integers and ra- 
tional fractions, 

To the foregoing principal tables are fubjoined feveral other fmaller 
tables, or fhort f|)ecimens of large ones : as, a table of the fines, tan- 
gents and fecants for every fingle degree of the quadrant, with tho 
correfponding lengths of the arc$, the radius being ioo,ooo,qk>o; ano-^ 
ther table of the fines, tangents and fecants, for ^ch degree aifo, ex« 
prefled in fexagefimal parts of the radius as far as the ^ order of parts } 
alfo two other tables for the m«UtipI|cationan4r^du£tion of fexagefimal 
quantities* 

The fecond part of this volume is intituled ll^iverfalhm InfpeSiio* 
nttm ad Canonem MatbtmatUum Libir fwgularis. It contains the con- 
^rudVion of the tables, a compendious treatife oji plane and fpherical 
trigonometry, with the application of them to a great variety of curious 
iuDje£ls in geometry and menfuration, treated in a very learned man<t 
ner ; as alfo many carious obfervations concerning the quadrafare of 
the circle, the duplication of the cube^ he. Computations are here 
eiven of the ratio of the diameter of a circle to the circumference, and 
of the length of the fine of i minute, both to many places of figures; by 
which he found thatthe fme of i minute is between 2,908,881,959 

and 2,908^882,056; alfo 
that, the diametier of a circle being 1000 &c, the perimeter of the in- 
fcribed and circumfcribed polygon of 393216 fides, will be as follows^ 

perim^ 



TRlGONdMETltlCAL TABLES, fcc^ 



perim. of ^hc infcrib. polygon 3^^^S9A^5%3S 
perim. of the circum. polygon 3X4>i599a65,37 
t^d that therefore the circymference cf the ckcle lies between thofd 
ijwo numbers. 

Although no author's name appears to the volyme I have been de« 
Icribine, thf re can be no doubt of its being the performance oi Vieta i 
for, befides bearing evident marks of his maflerfy hanci« it is mentioned 
by hijnfeif in ieveral parts of his other works coileiSed by Schooten^ 
4nd in the preface to thofe works by Elzevir the printer of them ; aa 
alfo in M. MontuclaU Hiftoirt des A^tbinuUiquii^ which are the only 
notices I have ever feen or heard of concerning this book, the copies 
of which are to rare,, that I never faw one befides that which is in my u 
own pofleffion, nor ever met with any. other perfon at aU acquainted X 
with fuch a book. ' 

In the otfael- worksof Vieta,publifliedat Leydenin 1646 by Schooten,' 
as mentioned above, there are feveral other pieces relating to trigono* 
l^try, fome of which^ on account of their originality andimportance^ 
^re very deferving of particular notice in this place* And firft, the 
very excellent theorems, here firft of all given % our author, relating 
10 angular fe&ions, the geometrical dcmonftrations of which are fup- 
f lied by that ingenious geometrician Alexander Anderibn^ a native of 
Aberdeen. We find here theorems for the chords (and coniequentl/ 
lines) of the fums and differences of arcs \ and fiDr the chords of arcs 
that ^re in arithmetical progreflion, namely, that the firftor leaft chord 
is to the 2d^ as any one after the firfl i& to the fum of the two next leis 
and greater, for example as the ad to the fum of the ift and ri^ and as 
the 3d to the fum of the ad and 4th, and as the 4^1 to. the mm of the 
3d and 5th, &c ; i^ that the i ft and ad being given, all the reft ard 
found from them by one fubtradion and one proportion for each, in ' 
which the I ft and ad terms are conftantly the fame: next are given 
theorems for the chords of any multiples of a given arc or angle, as 
alfo the chords of their fupplements to. a femicircle, which are fimiiar 
to the 'fines and cofines of the multiples of given angles ; and the con« 
clufions from them are exprefied in this manner: ift that if ^ be the 
chord of the fupplement of a given arc tf, to the radius i, then the 
chords of the fupplements of the multiple arcs, will be as in the 

annexed table : where the author ob- 

ferves that the figns are alternately 

+ and — ; that the vertical columns 

of numeral coefficients to the terms of 

the chords, are the fevei^al orders of 

figurate numbers, which he calb tri* 

angular, pyramidal, t^'iaogulo'trian- 

gular, triangulo- pyramidal, &c. gint" 

rated in the ordinary way by coy^tinual 

additions; not indeed from unity f as in 

THE CENEX ATiOM OF POWERS, but beginning With the number 2; and 

that the oowersobferve aJ ways the fame progrefion : i^condiy, that ff the 

f hofd 0/ an arc tf be called i > and d the Q\md of thi double arc lay then 



la 



2a 

4^ 

' Oa 
f 7^ 






t 

4c* + t 






1 ■ *■ 



r?« 






7c 



n w»i 



trs»» 



ttlSt.ORY OF 



Arcs 


Chords. 




la 


I 




2a 


d 




3^ 


^ - t 




4a 


d^ - 2^ 




i: 


./♦ - 3./» -- t 




</5 - 4^/3 ^ 3^ 




7« 




— 2 


8^ 


- 4^ 


&c; 


• 






Chords ;ind Chords of Sup; 



Chord 

Sup. ch.= — C* 
Chord = -C^ 
Sup. ch.= 
Chord = 
Sup. ch.= 
Chord = 



+C* 
--C* 

-C7 + 



i 

- CC3 

+ 6C* 
7C^ 



+ i 
+ S^ 

-I4C* 



+ ^ 
+ 7C 



the chords of the feries of multiple 
aces, will be as in this table \ where 
fAe author remarks as before on the 
law of the powers, figns, and coef- 
ficients, thefe being the orders of 
figurate numbers, raifed from unity 
by continual additions, afterthe man' 
ner of the genejis of powers^ which ge- 
neration in that way he fpeaks of 
as a thing generally known^ but 
without giving any hint how the 
coefficients of the terms of any power may be found from one another 
only, and independent of thofe of any other power^ as it was after- 
wards, and firft of all I believe, done by Henry Briggs, about the year 
1600 : and 3dly> that if C be the chord of any arc a^ to the radius i, 
then the feries of the chords and fupplemental chords of the multiple 
arcS) will be thus ; 
where the values are 
aiteratelychordsand 
chords of the fup* 
plements of the arcs 
onthefameline, and 
file law of the pow 
ers and coefficients 
as before^ but every 
alternate couplet of 
lines having their iigns changed. 

Another curious theorem is added to the al^ve, for finding the funi 
of all thefe chords drawn in a femicircle, froim one end of the diameter 
to every point in the circumference, thofe points dividing the circum'fe* 
rence into any number of equal parts ; namely as the leaft chord is 
to the diameter, fo is the fum of the faidleail chord and diameter and 
greatefl chord, to double the fum of all the chords including the dia-^ 
meter as one of them* 

As the above theorems are chiefly adapted for the chords of multiple 
angles, a few problems and remarks are then added (wKether by Vieta 
or Anderfon does not clearly appear, but 1 think by the latter) concern* 
ing the application of them, to the feAion of angles into fubmultiples^ 
and thence to the computation of the chords or fmes, or a canon of 
triangles. The general precept for the angular fedions is this; feled 
one of the above equations adapted to the proper number of the k&ioni 
jn which will be concerned the powers of the unknown or required 
quantity, as high as the index of the ie£lion; and from this equation 
find that quantity by the known methods for the refolution of equ&* 
tions. Examples are given of three different fe6Hons, namely for 3, 5^ 
and 1 equal parts, the forms for which are refpedively thefe 

30-^ O . . . • ; . .,=^ 

5 C — sC^ + c^ -g 

where g is the chord of the^iven arc or angle, and C th6 required chord 
of the 3d| Sthy or 7th part of it* And it is ihewn geometrically that 

the 



TUlGONOMETRieAL TABLES, &c. 9 

!the firft of tBefe equafions has 2 real pofitive ro6ts, the fecond 3, arid 
the laft 4 ; alfo from the fame principles the relations of thefe roots 
are pointed out. 

The method then annexed for conftruftrhg the canon of fines from 
the foregoing theorems, is th»s : By dividing the radius in ottreme- 
'and- mean ratio, is obtained the fine of x8 degrees ; this qurnquifeded^ 
gives the fine of 5° 36': Again^ by trifedhsg the arc of 60S there is 
obtained the fine of 20*^5 this again trifefted gives that of 6° 40' 5 and 
this bifefted gives that of 3° 20': Then, by the theorem for the difFc* 
fence of two arcs, there will be found the fine of 16', the diiFerenee 
l)ctween 3® 36' and 3® 20'': I^aftly, by four fucceffive bifeAtons, wiH 
at length be found the fines of 8^, 4', 2 and 1'. This laft beihg found, 
the fin^s of its multiples, tnd again of thfe multiples of thefe multiples, 
Ac, throughout the ^adrant, are to be taken by the proper theorems 
liefore laid down« 

And the fame fubjed is ftill farther ptirfued «nd explained iii t^e trad 
containing the anfwer given by Vieta to the problem prapofed to the 
whole world by Adrianua Romanus. 

In the fame colledion of Vietafs works, from page 400 to 432,- is 

J^iven a conrpleat treatife on prafiical trigonometry, containing rules 
or refolving ali the cafes of plane and fpherical triangles, by the Can9B 
Matbematieus, or talkie of fines, tangents and fecants^ 

The next authors whofe tabdors in this way have been printed, ate 
Rheticusj Otho, and Pitifcus: to all of nirhom we chire very great im^ 
provements in trigonometry. 

George Jovchim Rheticus, profeiTor of mathematics in the uhiver^ 
4tty of Wurtemburg, and fometime pupil toCopernicds^died in 1576, 
in the 6Qih year ot his age. He conceived and executed the great 
defign of computing the triangular canon for every 10 feconds of tht 
qttadrant, to the radius (loooocoooooooooo) confifting of i followed 
^y 15 ciphers. The feries of fines which Rheticus computed to this 
radius, for every 10 feconds, and f0r every fingle fecond m the firft 
and |aft degree of the quadrant, was publiffaed in folio at Franefort 
1613 by Pitifcus, who liimfelf added a'few of the firft itnes computed 
to the radius looocoocooooooooooooooo. 

Bat the large work, or whde trigonometrical canon, coinputed by 
Rheticus, was publiihed in 1596 by Valentine Otho, matbematiciah 
to the Ekdoral Prince Palatine, This vaft work contains all the 
three feries for the whole canon of right-angled triangles (being fimi- 
hrr ro the fines, tangents and (ecantfi, by which names I fhall call them}, 
with ali the differences of the numbers, to the radius 1 0006000000. 

Prefixed to thefe tables are feveral books oti their conftrudion aiid 
vfe in pl«ie and fpherical trigonoinetryy &c. Of tbef^^ the firft 3 
sre by Rheticus him felfj namely, book the ift containing the demon- 
ftratioBS of g lemmas concerning the properties of certain lihes drawn 
in and about circles: the id book contains 10 propofitions relative tb 
the fines and cofines of arcs, together with thofe of their fums and dif- 
ftrences, their halves and doubles, &c. The 3d book teaches, in 13 pro^ 
pofitioDS, theconftru£tionofthe canon to the radius lOOOooQOOOOOOOOO* 
py {pOie of the common properties of geometry having determined the 

C finea 



10 HISTORYOP 

lines of a few principal arcs, as 30S 36®, &c, in the firft propofition 
by continual bifeAions, he finds the fines of various other arcs, down 
to 45 minutes. Then, in the 2d propofition, by the theorems for the Aims 
and differences of arcs, he finds all the fin^s and cofines, up to 90 
degrees, in a feries of arcs differingby i^ 30^ And, in the 3d propofition, 
by the continual addition of 45^ he obtains all the fines and cofines 
in the feries whofe common difference is 45^ In the 4th propofition, be- 
ginning with 45^ and continually bife£ling, he finds the finesand co- 
fines of the feries of half arcs till he arrives at the arc of 14'*^ 19**, the 
fine of which is found to be i, and its cofine 999999999909999. In the 

fth propofition are computed the fine and cofine of 70^' or half a minute, 
n the 6th and yth propofitions are computed the fines and cofines for 
every minute, from 1' to 45', as well as of many larger arcs. The 
8th propofition extends the computation for fingle minutes much far- 
ther. In propofition 9 and 10 arc computed the tangents and fecants for 
all arcs in the ieries whofe common difference is 45'; and thefe are 
deduced from the fines bf the fame arcs by one proportion for each. In 
the remaining three propofitions, ij, 12, 139 are computed the tan- 
gents and fecants for feveral fmall angles. And from all thefe primary 
fines^ tangents^ and fecants, the whole canon is deduced and com- 
pleated. 

The remaining books in this work, are by the editor Otho ; namely, 
a treatjfe, in one book, on right-angled plane triangles, the cafes of 
which are refol ved by the tables ;. then right-angled fpherical trigonome- 
try in four books ; next oblique fpherical trigonometry in five books } 
and lailly feveral other books, containing various fpherical problems* 

Next after the above are placed the tables themielves, containing, 
for every 10 feconds, the fines, tangents and fecants, with all the dif- 
ferences annexed to each, in a fmaller charaSer. The numbers how- 
ever are not called fines, tangents, and fecants, but, like Vieta's be« 
fore defcribed, they are confidered as reprefenting the fides of right- 
angled triangles, and titled accordingly. They arealfo in like man- 
ner divided into three feries, namely, according as the radius, or con (Ian t 
fide of the triangle, is made the hypotenufe, or the greater leg, or the 
lefs leg of the triangle. When the hypotenufe is made the conftant 
radius ioqoooooooo, the two columns of this cafe or feries, are called 
the perpendicular and bafe, which are our fine and cofine ; when the 
greater leg is the conflant radius, the two columns of this feries are 
titled hypotenufe, and perpendicular, which are our fecant and tan- 
gent ; and when the lefs leg is confiant, the two columns in this cafe 
are called hypotenufe and bafe ; which are our cofecant and cotangent. 
After this large canon is printed another fmaller table, which is faid 
to be the two columns of the third feries, or cofecants and cotangents, 
with their differences, but to 3 places of figures lefs, or to the radius 
looooooo. But I cannot difcover the reafon for adding this lefs ta- 
ble, even if it were correfi, which is very far from being the cafe, the 
numbers being uniformly erroneous, and different from the former 
through the greeted part of the (able, 

Tpwafd^ 



^ilGdlJOMETRICAL TABLES, &c. it 

Towards the clofe of the i6th century mariy perfons wrote on the 
fulifje£t of trigonometry, and the conftru£tian of the triangular canon. 
But, their writings being feldom printed till ihany years afterwards, it 
is not eafy to affign their order in refpefl: of time. I {hall therefore 
mention but a few of the principal authors, and that without pretend- 
ing to any great precifion on tjie fcore of chrbnological precedence. 

In 1 591 Philip Lanfbergius firft publiflied his Gtometria Triangu^ 
iorum in four book, with the canon of fines, tangents, and fecants ; 
a brief but very elegant work 5 the Whole being clearly explained : 
and it is perhaps the firft fet of tables titled with thofe words. The 
fines, tangents and fecants of the arcs to 45 degrees, with thofe o£ 
their complements, are each placed in adjacent columns, in a very 
c^ommodious manner, continued forwards and downwards to 45 de« 
grces,:and then returning backwards and upwards to 9O degrees : the 
radius is looooooo, and a fpecimen of the firft page of the table is as 
follows I 



6 

• 



1 
1 

3 
4 
5 

ST 


Sinus 1 


Tiinjj^ns II Secants 


60 



2909 

5818 

87*7 
11636 

H544 


iboooooo 

9999999 
9999998 



2909 
5818 

8727 
11636 
14544 


Infinitum. 
J4377466738 
17188731915 


lodooooo 
10000000 

lOOOOOOA 


Infinitum. 
34377468193 
17188734824 


9999996 

9999993 
9999989 


11459152994 
8594363048 
68754B8693 


10000004 
10000067 
10060011 


114591573?? 
8^94368866 

6875495966 


. < 


. _ 








1891 



Of this wot k, the fitft book treats of the magnitude and relations of 
fuch lines.a^ are confidered in isind about the circle, as the chords, fines, 
tangents, and (bcaiits. tn the fecond book is delivered the conftruc- 
tion of the trigonometrical canon, by means of the properties laid down 
in the firft book : After which follows the canon itfelf. And in the 
third and fourth books is (hewn the application of the table, in the 
refolufioh ofplan^and fpherical triangles — ^^Lanfberg, who was born 
in Zealand 1561, was many years a mlnifter of the gofpel, and died at 
Middleb^rg iil 1632^ n 

The trigonometry of Bartholomew Pitifcus wis firft publifhed at 
Frirtcfort in the year IS99* This is a very compleat work ; contain* 
!ng, befides the triangular canon, with its conftru(9ion and ufe in re« 
folving triangles, the application oC trigonometry to problems of fur- 
veyingj altimetry, archrteSure, geography'i dialling, and aftronomy. 
, The conftrudtion of the canon is vtry clearly defcribed : And,^in the 
third edition of tbe book in the year 1612, he boafts to have added, in 
this part, arithmetical rules for finding the chords of the^d, 5th, and 
other uneven parts of an arc, from the chord of that arc peine given ; 
faying that it had been heretofore thought impoffible to give fuch 
rules : But, after all, thofe hoafted methods are only the application of 
the dou"ble rule of Falfe-Pofition to the then known rules for ^nding 
the chords of multiple arcs ^ namely, making the fuppofition of fomi 

C 9 Aumb^ 



ir 



HI STO « Y 6r 



liumber for the required chord of a fubmylfciple of any giv«n arc^ Aeit 
from this afiumed number computing what will be thie chord of its* 
multiple arc, and which is to be compared wilth that of the given arc ;- 
then the fame operation is performed with another fuppofition ; and (o^ 
on as in the douUe rule of pofition. The canon contains the fine, 
tangent, and fecant for every minute of the quadrant, in fome parts 
to 7 places of figures, in others to 8 ; as alfo idle differences for every 
Xo feconds. The fines, tangents, and fecants are alfo given for every 
jofeconds in the firfl and laft degree of the quadrant, for every 2 t^ 
conds in the iirfl and laft la minutes, and for every fingle fecond in 
die firft ^nd laft minute. In. this table the fines, tangents and iibcants 
are continued downwards on the leftrhand pages as for as to 45 de- 
grees, and then returned upwards on the rig^hc-jiand pa8;es, fo thafr 
the complements are always on the fame line in she oppoUte or facing 
pages. 

The mathematical works of Chriftopher Ckvius (a German jefuit, 
who was born at Bamberg in 1537) in five large folio volumes, were 
printed at Moguntia, or Meta, in 1612, the year in which the author 
d^ed, at the age pf 75* In the firft volume we find a very ^mple and- 
cfrcumftantial treatife on trigoqometry, with Regiomonta^us's canons 
o(F fines for every minute, as alfo canons of tangents^ and fecams, 
each in a fepara)te table, to the radius lOooocxxD'i and in a forni. 
continued forwards all the way up to 90 degrees^ The ejcplanation 
oif the conftru£tion of the tables, is very compleat, and is chiefly ex- 
tracted from Ptolemy, Purbach, and Regiomontanus. The fines 
have tire differences fet down foE each fecond, that is, the quotients 
arifing from the difiereiMes of the {xaes divided by 6o« About the 
year 1600 Ludolph van Cojleo, or a Ceulen, a refpedable Dutch* 
mathematician, wrote his book de circuh ^ adfirtptis^ in which he 
treats fully and ably of the pro|)ertics of line^ drawn in and ab^out the 
<;irqle, and efjpecially of chords or fubtenfes, with the confirudion of 
the canon or fines. The geometrical properties from which thefe 
lines are computed, arc the fame as thufe ufed by former writers \ but 
bis mode of computing and exprefling them, is difFerent from theirs f 
£or they a,dual]y extraded all the roots, &c, at every ftep, or Tingle 
operation, in decimal numbers; but he retained the radical expref- 
£oas to the laft, making them however always as fimple as poflible : 
^us, for ihftance, he determines the ildes of the polygons of 4^ 8, 
16, 32, &c, fides infcribed in the circle whofe radius is 1, to be as 
in the table annexed: 

\vhere the point before any figure 
(4/. 2) fignifies the root of all 
that follows it ; fo the laA line 
is in our notation the {^umt as 

V^2 — v^ + v^2 — -v/^ • And as 
the perfeS'management of fuch 

furds was then not generally known, he added a very neat tra<fl on 
that fubjeft. to facilitate the computations. Thefe, together with 
ptbcr difiertations on finiilar geometrical matters, were tranflated from 

the 



No.ot 
fides 

\ 

16 
&c 


Length of each fide.. 

iV/.2 — v'.2 4- v'* 



' tRlGfONOMEtRICAL TABLES, &c. jj 

dte Dutch language into Lttin by Wfllebrord SneH, aad puUlih^d at 
(Lugd. Batav?) Leydeo fn 1619. It was in this work that Ludolph 
determined the ratio of the diameter to thi circumference of the circle 
to 36 figures^ (hewing that, if the diametejr be i» the circuinference 

Witi be 

greater than 3' 141 S9>3t65 35,89793,23846,26433^3^79,50288^ 

but lefs than ri4l"59f 26535,89793,23846^26433,83279,50289; 
which ratio was oy his oriij^r* in imixation of Accbimedes, engravett 
on his tomb-ftone, as is witneflTed by the fald Snell, pa* 54, 55, Gychtm^p^ 
iricus, publiihed at Leydeo two years after, in ni^ich he treats the 
fajne fubj^d in a fimilar manner, recomputing and verifying Lu- 
dolph^s numbers. And in the &meix>ok he alfb gives a variety of 
geometrical approximations^ or n^echanical ipljutiojis, to determine 
very nearly the lengths of arcs, and the areas of foftors and Tegmenta 
of circles. 

Befides the Cyclomitncus, and another gtsometrical woik (jfpcJIanita 
BaUavui) publifhed in 1608, the fame Snellxoa wrote alfo foiir otberir 
th^rina trianguhrum tanomca^ in which ar£ contained the canoa of 
iecants, and in which the coajftru^tion ol fines, tangents and ie^ 
cants, together with the dimenuon or calculation of triangles, hotli 
p)aae and fpherica]^ are bxiefiy and clearly treated. After thr author's 
death this work was publiflied in 8vo, it Leydeu 1627, by Martimia 
llprtei^fius, who added to it a tatzSk on furveying and fpherica! piv>b- 
tems* Willebrord Snell was born in 1591 atKoyen, and died in 1^27^ 
being only 35 years^ of age. He was profefibr of mathematics in the 
ttniveriity of Leyden, as was alfo his father Rodolph Snetl. 

Alfi> in 1627, Francis van Scfaooten publifhed at Amfterdam, in a 
fenHl neat form, tables of £nes, tangents and Decants for every xnioute 
of the quadrant, to 7 places of figures, the radius being lOCQOOOO^ 
tc^echer with their ufe in the trigonometry of plane .triandes. Thefe 
SaUes have a great chara£her for their accuracy, being 4/eclared by th^ 
author to be without one fingle error. This however muft not br 
siadeiftood of the daft figure of the numbers, which I find to be very 
often ierroneous, fometimes in excefs and fometimes in defed, by not 
being always fet down to the neareft unit. Schooten died in 1659^ 
whife the lecond volume of his fecond edition of Defcartses' geomietry 
was in the prefs. He was alfo author of leveral other valuable works 
in geometry and other branches of the mathematics. 

The foregping are the prindpal writers oh the taUes of lines, tan- 
gents and fecants, before the invention of logarithms, which happened 
about this time, nameiy, foon after the year 160O4 Tables of the na^ 
tural numbers were now aJl compleated, and the methods of compu* 
ting them nearly perfe6ied : And therefore, before enterinjj; on the 
dilcovery and conftrv^tion of logarithms, I fiiall ftop here awhile ta 
give a fummary of the manner in which the faid natural fines, tan«' 
gents and fecsuits were a&ualLy computed, after having been gradu* 
ally improved from Hipparchus, Meoelau;, and Ptokmy, ndio u&d 
only tj^ .chords, down to the beginning ^f the i^th century, when 
fines, tangents, fecants and verfed fines were in ufe, and when the 
loetfaod hitherto employed had received its utmoft improvement. 

In 



54 HISTORTOF 

In this explanation I ihall here firft enumerate the theorems hf 
ixrhich the calculations were made, and then defcribe the applicatioti 
of them to the computation itfelf. 

Theorem I. The fquare of the diameter of a cirdei is equal to thef 
fum of the fquares of the chord of an arc and of the chord of its fup-< 
plement to a~femicircle. 

2.' The redtangle under the tvrai diagonals of an^ quadrilateral in- 
icribed in a circle^ is equal to the fum of the two redangles under the 
oppoiife fides; 

3« The fum of the fquares of the fine and cofine (hitherto called 
the fine of the complement), is equal to the fquare of the radius. 

4. The difference between the fines of two arcs that are equally 
diftant from 66 degrees, or i of the whole circumference, the one as 
much greater as the other is lefs, is equal to the fine of half th6 
difference of thofe arcs, or of the difference between either arc and the 
laid arc of 60 degrees. 

5. The fum of the cofine and verfed fine is equal to the radius^ 

6* The fum of the fquares of the fine and" verfed fine, is equal to 
the fquare of the chord, or to the fquare of double the fine of half' 
the arc. 

7* The fine is a mean proportional between half the radius and the 
Verfed fine of double the ate. 

8. A mean proportional between the verfed fine and half the radiu89 
is equal to the fine of half the arc. 

9. As radius is to the fine, fo is twice the cofine to the fine of 
twice the arc. . 

10. As the chord of an arc is to the fum of the chords of the 
fingle and double arc, fo is the difference of thofe chords to the chord 
of thrice the arc. 

1 1. As the chord of an arc is to the fum of the chords of twice and 
thrice the arc, fo is the difference of thofe chords to the chord of five 
times the arcs. 

12. And in general, as the^hord of an arc is to the fum of th^ 
chords of n times and n + i times the arc, fo is the diiferen^e of thofe 
chords to the chord of 2»+ 1 times the arc. 

13. The fine of the fum of two arcs, is equal to the fum of th6 
proau£ls of the fine of each multiplied by the cofine of the other and 
divided by the radius. 

14. The fine of the difference of two arcs, is equal to the difference 
of the faid two produAs divided by radius. 

15. The cofine of the fum of two arcs, isequal to the difference 
between the products of their fines and of their cofines divided by 
radius. 

16. The cofine of the difference of two arcs, is equal to the fum 
of the faid produds divided by radius. 

17. A fmall arc is equal to its chord or fine, nearly. 
i8. As cofine is to fine^ fo. is radius to tangent. 

19. Radius is a mean proportional between the tangent and co* 
tahgent. 

20. Half 



TRIGONOMETRICAL TABLES, &c. ,5 

20. H^lf the difference between the tangent and cotangent or an 
arc, is equal Jo the tangent of the difference between the arc and its 
complement. Or, the fum arifing from the addjtion of double the 
tangent of an arc with the tangent of half its complemeht, is equ^l 
to the tangent of the fum of that arc and the iaid half complement. 

21. The fquare of the fecant of ^n arc, is equal to the fum of the 
fquajres of the radius and tangent. 

22. Radius is a mean proportional between the fecant and cpftn^ 
Or, as cofine is to radtps, fo is radius to fecant, 

23. Radius is a mean proportional between the fine and cofecant. 

24. The fecant of an arc, is equal to the fum of its tangent and the 
tangent of half its complement. Or, the fecant of the difference be- 
tween an arc and its complement, is equal to the tangent of the fai4 
difference added to the tangent of the lefs arc. 

25. The fecant of an arc, is equal to the difference between the 
tangent of that arc and the tangent of the arc added to half its comple- 
ment. Or, the fecant of the difference between an arc and its com- 
plement, is equal to the difference between the tangent of th^ faid dif- 
ference and the tangenjtt)f the greater arc. 

From fome of thefe 25 theorems, extracted from the writers before 
mentioned, and a few propofitions of Euclid's elements, they compiled 
the whole table of fines, tangents, and fecants, nearly in the following 
manner. 

By the elements were computed the fides of a few of the regular 
figures infcribed in a circle, which were the chords of fuch parts of 
the whole circumfei^nce as are expreffed by the number of fides, and 
therefore the halves of thofe chords the fines of the halves of the arcs. 
So, if the radius be loooocoo, the fides of the following figures wiii 
give the annexed chords and fines. 



The figure 



Triangle 

Square 

Pentagon 

Hexagon 

Decagon 

Quindecapon 



Arcs fub- 
tended 



120'' 

60 

36 

24 



its chord, 
or fide 



Half 
arc 



17320508 
I4i4?i36 
11755705 
I 0000000 
61B0340 

4158234 12 



60^ 

4'> 
36 

30 

18 



Its fines, 
or i chord • 



8660254 
7071068 
5877853 
5000000 
3090170 
2079117I 



Of fome, or all of thefe, the fines of the halves were continually 
taken, by theorem the 6th, 7th, or 8th, and of their complements by 
the 7d ; then the fines of the halves of thefe, and of their complements 
by the fame theorems ; and fo on alternately of the halves and com- 
plements, till we arrive at an arc which is nearly equal to its fine. 
Thus, beginning with the above arc of 12 degrees, and ijs fine, we 
jybt^in the halves as follows f - ' 

Tl»9 



t6 



HISTORY OF 



F 



he halves 

i 

I 3d 

45 



The c6nip. 
of thefe 

II 

89 



Their finesi 
104528$ 

261769 
1308^6 



50 
15 



The halves 
of thefe 

21 
16 



4? 

21 

44 



30 

45 
*5 



99+5218 
9986295 
9996573 
9999143 



6691306 

3583679 
1822355 

91^616 

6883545 

3705574 
697790J 



Thp comp. 
' of thefe 

48* / 

69 

79 
84 

^6 
68 

45 



4$ 
30 

15 
45 



Sines 

7431448 
9335804 
9832549I 

9958049 

7^53744 
9288095 

7163019 



^The halves 
of ^efe 




4067366 
566^062 
296^416 
6394390 

3947439 



The halves 

33 ' 
16 30 

8 15 

J7 4£ 

Coinps. 



Sines 

544639b 

I2840153! 

1454926 

4656745 



57 

7J 
&z 

62 



30 
45 
15 



lialvts 
28 30 
14 15 

36 45 



'^35455 
241262 

95C0199. 

7688418 

9187912 



I 



' Comps. 
61 30 

75 45 
55 ^5 



8386706 
9588197 
98965)^4 
8849876 



4771 588 
2461533 
5983246 



8788171 
9692309 
8012538 



Half 

30 45 



Comp. 
59V 'S 



511293J 



-r 



8594064 



The fines of fmall arcs are then deduced in this manner. Front 
tbe fine of 43^ above determined^ ate found the halves^ which will t>Q 
ffaus; 

4S' o" . . . . • 130896 

aa 30 65449,4 

II 15 . . • . . 32724,8 

Now thefe laft two fines being evidently in the fztne ratio a« their 
arcs^ the fines of all the lefs fingle minutes will be found by fingle 
proportion. So the 45tli part of the fine of 45^ gives 2909 for the fine 
of i' ; which may be doubled, tripled, &c, for the fines of 2', 3', &c, 
up to 45'. 

Then, from all tht foregoing primary fines, by the theorems for 
halving, doubling, or tripfrng, and by thofe for the fums and diffe-« 
rences, the reft of the firrei are deduted, to corhpleat the quadrant. 

But having thtts determined the fines and cofines of the firft 30® of 
the quadrant, that is the fines of the firft and laft 30S thofe of the 
intermediate 30^ are, by theor. 4, found by one fingle fubtra^ion fo( 
eath fine. 

Tiie fines of the whole quadrant bring thus: ^ompleated, the tarr- 
gents arefeondby theor. 18, l^f 2(3, hamfcly for one half of the quad- 
rant by the i8th and 19th, and the dther half, by one fingle addition 
.or fui>tra£tion for each, by the ickh theorem. 

And laftly, by theor. 24 and 15^ the fecants are deduced froni the 
tangents by addition and fubtradion only. 

Among the various means ufed for confi:ru£ting the cahon of fines, 
tangents and fecants, the writers above enumerated feem not to have 
been poflTefied of the method of diflferences, fo profitably ufed fince. 
and firft of all I believe by Briggs, in computing his trigonometrical 

canon 



TRrOONOMErHICAL TABLES, «cc. 17 

tahon dnd his logarithms, as we fhall fee hereafter when we come to 
defcribe thofc works. They took however the fuccefSve difFerences of 
the numbers after they were computed, to verify or prove the truth pf 
them; and if found erroneous, by any irregularity in the laft diiFerencesy 
from thence they had a method of corre£^ing the original numbers 
themfelves. At lead this method is ufed by Pitifcus, Trig, lib, a, where 
the difFcrentes are extended to the third order.— In ^a. 44, of the 
fame book alfo is defcribed, for the firft time that I know of, the common 
notation of decimal fradions as now ufed. And this fame notation 
was afterwards defcribed and ufed by baron Neper in pojitio^ and 5 of 
his pofthumous work on the conftrudtion of logarithms, publifhed by 
his Ion in the year 1619* But the decimal fra6lions themfelves may 
be confidered as having been introduced by Regiomontanus, by his de- 
cimal divifion of the radius &c. of the circle ; and fronx that time gra-v 
dually brought into ufe; but continued long to be denoted after the 
manner of vulgar fradions, by a line drawn between the numerator 
and denominator, which laft however was foon omitted, and only the 
numerator fet down with the line below it; thus it was firft 3It-cIi 
the 31 -U^; afterwards omitting the line it became 31^% and laftly 
3I3, or 3i»35 or 3i'3S : As may be traced in the works of Victa and 
others fince his time, gradually into the prefent century. 

Having often heard it remarked that the word J!ne, or in Latin and 
Frenchjinusj is of doubtful origin ; and as the various accounts which 
I have fecn of its derivation, are very different from one another, it 
fliay not be amifs here to employ a few lines on this matter. Some 
authors fay this is an Arabic word, others that it is the fingle Latin 
ViovAJinusj and in Montucla*s Hijlnndes Mathematiquet^ it is conjee** 
tured to jl>e an abbreviation of two Latin words. The conjecture is 
thus expreifed by the ingenious and learned author of ^hat excellent 
hiftory, at pa. xxxiii among the additions and correftions of the firft 
volume : ** A I'occafion des finus dont on parle dans cette page, coma 
d'une invention des Arabes, voici uneetymologie de ce nom, tout-a-faic 
heureufe & vraifemblable. Je la dois a M. Godin, de I'Academie 
Royale des Sciences, Dire£teur de TEcole de Marine de Cadix. Les 
finus font, comme Ton fcait, des moities de cords; h les cordes en 
Latin fe nomment infcripta, Les finus font done ftmiJIis infcriptarunif 
ce que probablement on ecrivit ainfi pour abr^ger, S. Ins. Dela en- 
fuite s'eft fait par abus le mot de finus.'* Now ingenious as this 
conjedure is, there appears to be little or no probability for the truth 
of it. For, in the fixft place, it is*not in the leaft fupported by quo- 
tations from any of the more early books to (hew that it ever was thq 
pradice to write or print the words thus 5. Im^ upon, which the con- 
jeAure is founded. Again, it is faid the chords are called in Latin 
infcripia\ and it is true that they ibmetimes are lb; but I think they 
are more frequently cMed fubtinfa^ and the &nt% femijfis fubtenfarum 
of the double arcs, which will not abbreviate into the word Jinus. 
But it may belaid, whatreafon have we to fuppofe this word to be either 
a Latin word, orthe abbreviation of any Latin words whatever? that 
it fe6nis but proper to feek for the etymolcgy of words in the language 

D of 



:i8 H I S T O R T O F 

of the Inventors of the thingi. For which reafon it \^ that we £n<£ the 
two other words, tangens Mi/ecans^ are Latin, as they were invented 
and ufed by authors who wrote in that language. But the (rne^ are 
acknowledged to have been invented and introduced by the Arabians^ 
and tbenCe by analog it would feem probable that this is a word of 
thiir language, and from them adopted, tbgether with the u(e of it* 
by the Europeans-. And indeed Lanfbergius, in the 2d pa. of his tri- 
gonometry" above -meationed, expreiHy fay3 that it ix Arabic: Hi» 
words are^ Voxjinus Arabica efiy et proinde harbard ; fid cum longa ufw 
mpfrahatafit^ ^ cemmodior mnjiippetatrnequaquam repudienda efi :factln 
$ntm in verbis nos ijfi opertetj cum de rebus cenvenit. And Vieta fays 
ibmething to the fame purport in pa. 9 of his UniverJaHum InfpeSHonum ad 
Canonem MdAematicum Liber : His words are, 6revejinm vocabulum^ 
ciim Jitartisj Saracenis prsefirtim quam familiarey non ejt ab artificibms 
gxplodendumj ad laterumfimijjmm infiriptcrum demtationemy tf ^ . 

Guarinus alfo is of the fameoptnion : in his EucUdes Adau^fus &$. tra^ 
, XX, pa. 307. he fays. Sinus vero eft ncmen Arabicum ufurpaium in hone 
Jignificationem a matBematteis ; although be was aware that a Latin ori* 
sin was afcribed to it by Vitalis, for he immediately adds, Liset Vitalise 
tnfuo Lexu9 Mathematics ix eo yeUtJinum appeileUumj quid claudat cur^ 
vitafem arcus* 

Long before I either faw or heard of any conjecture or obfervation* 
concerning the etymology of the wdrd^narj, I remember that I ima^ 

fined it to be taken from the fame Latin wprd, fignifying breaft or 
ofom, and that our fine was (o called aIlegorica]ly» I had obferved 
that feveral of theternis in trigonon^try were derived from a bow to 
flioot withy and its appendage?'; as arrtr^the bow, chorda the ftring, 
zndfifgifta the arrow, by which name the ver&d fine, which* repre-* 
fents it, was fometimes called ; alfo that the tangens- was fo called 
*from its office, bein^ a line touching the circle,. and y^r^xrx from its cufi' 
ting the fame ; I therefore imagined that tho^mus was fo called, either 
from its refemblance to the breaft or bofom, or. from its beina; a line 
drawn within the bofbm (ftnus) of the arc, or front its being that part 
of the ftring (chorda) of a bow (arcus) which is drawn near the breait 
(Jinus) in the a£): of (hooting. And perhaps Vitalis's definition above- 
quoted hasibme allufion to the iame fimilitude. 

Alfo Vieta feems to allude to the fame thing in calling finus an alle^ 
gorical word, in pa. 417 of his works as publimed by Scbooten, where, 
with his ufual judgment and precifion, he treats of the propriety of the 
terms ufed in trigonometry for certain lines drawn in and about the- 
circle, of which, as it very well deferves, I (hall here extraft the pria- 
cipal part, to fhew the opinion and arguments of fo great a man oi^ 
thofe names. '< Arabes autem femiffts infcriptas dupio, numerispra^* 
fertim acftimatas, vocaverunt allegorice Sinus, atque.ideo ipfam fenai- 
diametrum, quae maxima eft femiffium infcriptarum, Sinum Totum* 
£t de lis fua methodo canones exataverunt qui clrcumferuntur, fuppu^ 
tante praefertim Regiomontano bene jufte & accurate, in iisetiam pat^ 
(iculis qualium femidiameter adfumitur io,ooo,ooo» 

« Ex 



TRIGONOMETRICAL TABLES, «£c €9 

** Ex canonibus deinde finuum derivaverunt recentiores canonem 
femiffium circuixifcriptarum/ quern dixere Fcecundum ; & canonem 
educarum e centro, quern dixere Foecundiinmum & Beneficum, 
hjrpotenufis zddi&um. Atque adeo femifTes circumfcriptas, nu* 
jneris praefertim acftimatas, vocaverunt Fcecundos Sinus nuraerofve 
videlicet; quanquam nihil vetat Fcecundi nomen fubftantive accipi. 
Hypotenufas autem Beneficas, vel etiam fimpliclter Hypoten^fas: 
quoniam hypotenufa in prima feHe finus totius nomen retinet. Ita* 
que ne novitate verborum res adumbretur, & alioqui fua arcificibus 
eo nomine defaita prxripratiir gloria, prspofitain Canone Machematico 
canonicts numeris inicriptio, candide admonet primam ferlem efle 
Canonem Sinum. In lecunda vero, pnrtem canonis foecundi, partem 
canonis foecundiflimi, contineri. In teitia, reliquam. 

Sane practer infcriptas & clFCumfcfiptaSp circulum etiam adficiunt 
jfliae lineae reftae, velut Incidentes, 7'angentes, & Secantes. Verum 
illae voces fubftantivae funt, non peripheriarum relativae. Ac fecare 
<[uidem circulum linea reda tunc intelligitur, cum in duobus pundis 
fecat. Itaque non loquuntur bene geometrice, qui edudas e centro 
ad metas circumrcriptarum vocant fecantes improprie, cum fecantes, 
& tangentes ad certos angulos vel peripherias referunt« Immo vero 
artem confundunt, cum his vocibus necefle habeat uti geometra abs 
f^Iatione. 

** Quare fi quibus af rideat Arabum metaoiiora ; quae quidem aut 
omnino retinenda videtur, aut onmino explodenda ; .ut femiiles in* 
icriptas, Afabes vocant finus ; fie femiffes circumfcriptae, vocentur 
Profinus Amfinufve ; & cd\Jt6kx e centro, TranfSnuofae. Sin allego- 
Ha difpliceat, geometr'rca fane infcriptarum & circumfcriptarum no* 
mioa retijieantur. £t cum <du£tae e centro ad metas circumfcripta- 
rum, non habeant hadenus nomen certum neque elegans, voceantur 
fiine proTemidiametri, quafi protenfse femidiametri, febabentes ad fuas 
circumfcriptas, ficut lemidiametrr ad infcriptas/' 

Againft the Arabic origine however of this word (Jinus) may be ur- 
ged its being varied according to the fourth declenfion of Latin nouns^ 
\\k!tmanus'^ and that if \t were an Arabic word latinized, it would 
have been ranked under eitiier tht firft, fecond, or third declenfion, jas 
is ufual' in Atcb adopted words. 

So that, upon the whole, it will perhaps rather feem probable, that 
the ttxmfoius is the Latin word anfwering to the name by which the 
Saracens called that line, and not their word itfelf. And this conjec- 
ture leems to be rendered fiill more probable by fome expreffions in 
pa. 4 and 5 of Otho*s preface to Rheticus's Canon, where it is not only 
iaid^ that the Saracens called the half chord of double the arc finus^ 
but alio that they called the part of the radius lying between the fine 
and thearc;^«/ vsrfus vel ftgttta^ which are evidently Latin words, 
and feem to be intended for the Latin tranflations of the names by 
which the Arabians called thefe lincs^ or the numbers expreifing the 
lengths of tbem« 

T>% OF 



[ 20 3 



OF LOGARITHM S. 

t 

THE trigonpmetrical canon of natural ftnes, tangents and fecants» 
being now brought to a condderable degree of pcrfc£^ion ; the 
great length and accuracy of the numbers, together with the increaf- 
jng delicacy and number of aftronomical problems and fpherical tri- 
angles, to the refolution of which the canon was applied, urged many 
perfons, converfant in thofe matters, to endeavour to difcover fome 
means of diminifhing the great labour and tfme, requifite for fo many 
multiplications and divifions, in fuch large numbers as the tables then 
confiited of. And their chief aim was, to reduce the multiplications 
and divifions to additions and fubtra£lions, as much as poffible. 

^For this purpofe, Nicholas Raymer Urfus Dithmarfus invented an 
ingenious method, which ferves for one cafe in the fines, namely^ 
when radius is the firft term in the proportion, and the fines of two arcs 
are the fecond and third terms; forhefhewed that the fourth term or fine» 
would be found by only taking half the fum or difference of the fines 
of two other arcs, which (hould be the fum and difference of the lefs 
of the two former siven arcs and the complement of the greater. This 
is no more in effeS than the following well-known theorem in trigo- 
nometry: As half radius is to the fine of one arc, fo is the fine of another 
arc to the cofine of the difference minus the cofine of the fum of the 
faid arcs. The author publifhed this ingenious device in 1588) in his 
FundamentumJJironomi^. And three or four years afterwards it was greatly 
improved by Clavius, who adapted it to all proportions in the refolu- 
tion of fpherical triangles, both for fines, tangents, fecants, verfed fines, 
&c; and that whether radius be in the proportion or not. All which 
he explains very fully in km. 53 lib. i. of his treatife on thtjtfftrolabcn 
This method, although ingenious, depends not on any abftrad property 
of numbers, but only on the relations of certain lines drawn in and 
about the circle, and it was therefore rather limited^ and fometime^ 
attended with trouble in the application. 

After perhaps various other contrivances, inceffant endeavours tX 
length produced the happy invention of logarithms, which are of di-v 
re£k and univerfal application to all numbers abftradcdlv confidered, 
being derived from a property inherent in themfelves, 1 his property 
may be confidered, either as the relation between a geometrical . feries 
of terms and a correfponding arithmetical one, or as the relation be« 
tween ratios and the meafurcs of ratios, which comes to much the fame 
thing, they having been conceived in one of thefe ways by fome of the 
writers on this fubjc A, and in the other by the reft of them, as well as 
in both ways at different times by the fame writer. A fummary idea 
of this property, and of the probable reflcftions made on it by the firft 
writers on logarithms, may be to the following effe£t. 

The learned calculators, about the clofe of the 16th, and begin- 
ning of the 17th century, finding the operations of multiplication and 
divifion by very long numbers cf ^ or 8 places of figures, whic^ they 



LOGARITHMS. %t 

had Frequently occafion to perform in folving problems relating to 
geography and aftronomy, to be exceedingly troublefome, fet-them* 
felves to condder whether it was not poffible to find fome method of 
ieflenine this labour, by. fubftituting other eafter operations in their 
ftead. In purAiit of this objeft they reflected that, fince in cverjr 
multiplication by a whole number, the ratio^ or proportion, of the 
produd to the multiplicand, is the fame as the ratio of the multiplier 
to unity, it will follow that the ratio of the produd to unity Cwhich^ 
according to Euclid^s definitiqp of compound ratios, is compounded 
of the ratios of the faid product to the multiplicand and of the multi- 
plier to unity), muft be equal to the fum of the two ratios of the mul- 
tiplier to unity and of the multiplicand to unity. Confec^uently, if 
they could find a fet of artificial numbers that fhould be the reprefenta- 
tives of, or (hould be proportional to, the ratios of all forts of numbers 
to unity, the addition of the two artificial numbers that fliould repre- 
fent the ratios of any multiplier and multiplicand to unity, would an« 
fwer to the multiplication of the faid multiplicand by the faid multi-> 
plier, or the fum arifmg from the addition of the faid reprefentative 
numbers, would be the reprefentative number of the ratio of the pro- 
4]ud to unity ; and confequently the natural number to which it Ihould 
be found, in the table of the faid artificial or reprefentative numbers^ 
that the faid fum belonged, would be the produ^ of the faid multipli- 
cand and multiplier* Having fettled this principle as the foundation 
of their wiflied-for method of abridging the labour of calculations^ 
they refolved to tompofe a table of fuch artificial numbers, or num-> 
bers that fhould be reprefentatives of, or proportional to, the ratios of 
all the common or natural numbers to unity. 

The firft obfervation that naturally occurred to them in the purfuit 
cf this fcheme, was that, whatever artificial numbers fhould bechofen 
to reprefent the ratios of other whole numbers to unity, the ratio of 
equality, or of unity to unity, mufl be repreiented by o \ becauie that 
ratio has properly no magnitude, fince, when it is added to, or fub* 
traded from, any other ratio, it neither increafes nor dimini(hes it* 

The fecond oofervation that occurred to them was, that any num- 
ber whatever might be chofen at pleafure for the reprefentative of the 
ratio of any given natural number to unity ; but that^ when once fuch 
choice was made, ail the other reprefentative numbers would be 
thereby determined, becaufe they muft be greater or lefs than that firfl 
reprefentative number, in the fame proportions in which the ratios re- 
preiented by them, or the ratios of the correfponding natural numbers 
to unity, were greater or lefs than the ratio of the laid given natural 
number to unity* Thus, either i, or 2, or 3, &c, might be chofen 
for the reprefentative of the ratio of 10 to i. But, if 1 be chofen for it, 
the reprefentatives of the ratios of 100 to i and 1000 to i, which are 
double and triple of the ratio of 10 to i, mufl^be 2 and 3, and cannot 
be any other numbers ; and, if 2 be chofen for it, the reprefentatives 
of the ratios of 100 to i and jooo to i will be 4 and 6, and cannot be 
jiny other numbers ; and^ if 3 be chofen for it) tb^ reprefentatives of 

the 



»t HI3TORTOF 

t^ ratios ofioo to i an4 lOOOto i will be 6 and 9^ and cannot be uny 
ptber numbers ; and fo on. 

The third obfervation that occurred to them was, that, as thefe 
artificial numbers were Kprefent^tives of, or proportional t<s ratios 
of the natural numbers to unity, they muft be expreiBons of the num- 
bers of Tome fmaller equal ratios that are contained in the faid ratios. 
Thusy if I be taken for the reprefentative pf the ratio of 10 to i, then 
3, which is the reprefentative of the ratio of 1000 to I, will exprefs the 
number of ratios of 10 to i that are contained In the ratio 9f IQOO to i. 
And ifyinftead of i,wemakeiO9O0O|0OO,ortenmillions,thereprefenta« 
five of the ratio of 10 to i , (in which cafe i will be the repreientative of 
a very fmall ratio, or ratiuncuUty which is only the ten-millionth part 
of the ratio of 10 to i, or will be the reprefentativ<i of the xo,ocx>,oooth 
footofjOi or of the firft or fmalleft of 9,999,999 mean proportionals 
anterpofed between i and 10), the reprefentative of theratioof 1000 to 
I, which will in this cafe be ^,oop«ooo, will expref^ the number of 
tbofe miiuncul^tf or fmall ratios of the io,ooo,oooth coot of lo to i^ 
il^bich are contained in the faid ratio of 1000 to i. And the like may 
be (hewn of the reprefentative of the ratio of any other number t<i 
«inity« And therefore they thought thefe artificial nupnbers, which 
thus reprefent, or are proportional to, the magnitudes of the ratios of 
the natural numbers to unity, might not improperly be called the 
l^GARiTHMS of thofe ratios, fince they exprefs the numbers of fmal- 
ler ratios of which they are compofed. And then, for the fake of 
brevity, they called them the Logarithms of ihe [aid natural nnmiers 
AemfeivfSy which are the ^antecedents of the faid ratios to unity, of 
which they are in truth the reprefentativcs* 

The foregoing method of confidering this property, leads %o much 
the fame concluiions as the other way, in which the relations betweeo 
a geometrical iRnries of terms, and their exponents, or the terms of aa 
arithmetical (eries, are contemplated. In this latter way, it readily 
occurifed that the addition pf the terms of thp arithnpietical feries eor-i 
reiponded to the multiplication of the terms of the geometrical feries ; 
and that the arithmeticals would therefore form a fet 0f artificial num* 
bers, which, when arranged in tables with their gpeometrieals, wouhj 
'SUifwer the purpofes defired, as has been explained above. 

From this property, by afiuming four quantities^ two of them as 
two terms in a geometrical feries, and the others as the two corref** 
ponding terms of the arithmeticals, or artificials, or logarithms, it ie 
evident that ail the other terms of both the two feries may thence be 
generated. And therefore there aiay be as many <et$ or fcales of logar-^ 
ithms as we pleafe, fmce they depend intirely on the arbitrary afitimpiN 
tion.of the firft two arithm^ticals* And all poffible natural numbers 
may he fuppofed to coincide with fome of the terms of any geometrical 
progreifion whatever, the lo^rijthms or arlthmeticals determining 
jpyhich pf the terms in that progrefflon they are. 

It was proper however that the arithmetical feries ihould be fa af- 
fumed,, as that the term o in it might anfwer to the term x in the geo« 
metricals \ otherwile the fum of the logarithms of any two numbers 

would 



LOGAflirRM& 

wovid be zhnp to be dioiinifllied by tile logarithm of i^ to give the^ 
logarithm of the produd of thofe numbers : for which reafon, makiitg 
O the logarithm of i, and affuming any quantity whatei^er for the value 
of the logarithm of any one nummnr^ the logarithms of all other nwii^ 
bers were th«nce to be derivedl And bence^ like as the multipitcarti^i# 
of two numbers is eSe&ed by barely addii^ their logarithms, fo divt-* 
fion is performed by fubtra£ling the logarithm of the one from thal^ 
of the other, raifing of powers by mukiplying; tbe logarithm of Ac 
given number by the index of the power, and extradion of roots hf 
dividing the logarithm by the index of the root. It is alfo evident 
that, jh all fcates or iyftems of logarithms^ the logarithm of o will be 
infinite } nanasly, infinitely negative if the logarithms {ndretfe with* 
the natural numbers, but infinitely pofitive if the contrary ; becaufe 
that while the geometrical feries muft d^creafe through ifmnite divi- 
fions by the ratno of the progreffion, before the quotient come to a or 
nothing; the logarithms, or arithmeticaky will tn like manner uo- 
dergo the correfponding infinite fubtmftions or additions of the com- 
mon equal diflSsrence ; which equal increafe or decteafe, thos inde* 
finitely continued, muft needs tend to an infinite refult. 

This however was no newly difcovered property of numbers, but 
wha£ was always well known to all mathematicians^ bdng treated of 
in the writings of Euclid, asalfoby Archimedes, who nfeadie^ greait 
nfc of it in his AnnariMS^ or treatife on the number of the (hnds^r* 
namely, in affigninj; the rank or place of thofe terms, of a geome^^ 
trical feries, produced from the muitiplic^tion togethi^l^ of any of 
the foregoing terms, by the addition of the correfponding terms of 
the arithmetical feries, which ferved as the indices or eroponents of 
the former. And tbe reafon why tables of thefe numbers were not 
iboner compofed, was, that the accuracy and trouble of trigonome- 
trical computations had not fopner rendered them necellary* ft is 
therefore not to be doubted that, about the dole of the fixteenth and 
beginning of the feventeenth century, many perfons had thoughts of 
fuch a table of numbers, befides the few who are faid to have at« 
Sempted it. 

Longomontanus has, by fome, been faid to have invented logarithms: 
'but this cannot Well be fuppofed to have been much more than in 
idea, fince he never publtfhed any thing of the kind, nor ever iaidr 
claim to the invention, though he lived thirty^three years after they 
were firft publtfhed by baron Neper, as he died only in 1647, when' 
they had been long known and received all over Europe* Some cir* 
eumftances of this matter are indeed related by Wood in his jhhina 
Oxonienfis^ under the artitrle Briggs, on the authority of Oughtred 
and Wingate, viz. '^ That one Dr. Craig a Scotchman^ coming oufr 
of Denmark into his own country, called upon Joh, Neper baron of 
Marchefton near Edenburgh, and told him among other difcourfes of 
a new invention in Denmark (by Longomontanus as 'tis faid) to fave 
the tedious multiplication and divifion in aftronomical cakuhtion». 
Neper being folicitous to know farther of him concerning this matttr, 
be could give no other account of it> than that it was by proportion**. 

able 



HISTORY OF 

aUe numbers^ Which hint Neper takings be defired him at his re 
turn to call upon him aeain. Craig, after fome weeks had pailed^ 
dbd fo, and Neper then mewed him a rude draught of that he called. 
Canon mirabilis Logarithmorum^ Which draught, with fome altera- 
tions^ be printing in 1614^ it came forthwith into the hands of our 
author Briggs, and into thole of Will. Ougbtred^ from whom the 
relation of this matter came.'' 

Kepler alfo fays that one Jufte Byrge, af&ftant aftronomer to the 
landgrave of Hefie, invented, or proje£led, logarithms long before 
Neper did, but that they bad never come abroad on account of the 

freat relervednefs of their author with regard to his own compofttions^ 
»yrge is alio faid to have computed ,a table of natural fines for every 
two feconds of the quadrant.. 

But whatever may have been faid or conjeSured concerning an/ 
thing that may have been done by others, it is certain that the world 
is indebted, for the firft publication of logarithms, to John Napier, 
or Nepair ♦, or in Latin, Neper, baron of Mcpchifton, or Markinfton, 
in Scotland, who died the 3d of April 1618, at the age of 67 years. 
Saron Napier added confiderable improvements to trigonometry, and 
the frequent numerical computations he performed in this branch, 
gave occafion to his invention of logarithms, in order to fave part of - 
the trouble attending thofe calculations ; and for this reafon he 
adapted his tables peculiarly to trigonometrical ufes. 

This difcovery he publiihed in 161 4, in his'book intituled Mtrifici 
bgaritbmorum canonis defcriptio^ referving the conftrudion of the num- 
bers till the fenfe of the learned concerning his invention (hould be 
known. And, excepting the cdnftrudion, this is a perfect work on 



* The ongine of which name Crawfunl informs us was fzx)m a (lefs) peeri^/i action of 
ene of his anceftors, viz. Donald, fecond fon of the earl of Lenox in the time of David ther 
fecond. **• Some Engliih writers 1 ntiilaking the import of the term bcrofi^ having called 
this celebrated perfon lord Napier, a fcotch nobleman. He was not indeed a peep of 
Scotland ; but the peerage'of Scotland informs us, that he was of a very antieut, honour- 
able, and illuftrious family ; tliat his anceftors, for many generations, bad been poiTefied of 
fnndry baronies, and, amongft others, of the barotiy of Merchiftoun, which defcended to 
liim by the death of his father in 1608. Mr. Briggs, therefore, very properly- ftiles him 
Baro McrchiJfmtL Kow, according to Skene, ^ vcrbcrum Jknipcatlouc, * In this realm (of 
Scotland^ he is called an Barrone, quha haldis his landes immediatelie in chiefe of the 
king; ana hes power of Pit and Gallows; FoffaetFurca ; quhilk was firft inftitute and granted 
be king Malcolme, quha gave power to the Barrones to have ane Pit, qubairin wen>en 
condemned for thieft fuld be drowned, and ane Gallows, whereupon men thieves and. 
trefpaflowres, fuld be hanged, conforme to the doome given in the Ban'on Court there- 
anent.' So that a Scotch baron, though no peer, was ne%erthelefs a very confiderable 
perfona^e, both in dignity and power.*' Jitid's EJfay m Logar'ahviu — The name of the illuf- 
tiious inventor of logarithms, and his family, has been varioufly written at diil'erenc 
times, and on different occafions. In his own Latin works, and in (perhaps) all other 
books in Latin, it is Nepo; or Ntptna Baro Mtrcbijknii • By Briggs, in a letter to Archbi- 
Ihop Uiher, he is called J^aper^ lord ofMarkinJkn : In Wright's tranflation oi the logarithms, 
which wasrevifed by the author himfelf, and publiflied in 1616^ he is called Ntfair, tann 
of MarchtRm \ and the fame by Crawfurd and fome others : But M'Kenzy- and others 
write it ffapiert barw of Mcrcbifon ; which, being alfo the orthography now ufed by the 
family, I fhall adopt in this work. I obfenre alfo that the Scotch Compendium of Ho« 
now fays he was only Sir John Napier, and that his fon and heir, Archibald, was the iirft 
lord, being raifed to that dignity in 1626. Be this however as it may, I IhaU ednform 
Co the conuBOA modes of expre^^> and call him indifferently harm Napkr or MNapur^ 

this 



'LOGARITHMS. »$ 

■ 

tWs kind of logarithms, containing in efFeft the logarithms of all num- 
bers, and the logarithmic fines, tangents, and fecants for every minute 
of the quadrant, together with the tlcfcription and ufes of ihe tables, 
asalfo his definition and idea of logarithms. 

Napier explains his notion of logarithms by lines defcribed or ge- 
nerated by the motion of points, in this manner: He firft conceives a 
line to be; generated by the equable motion of a point, which pafies over 
equal portions of it in equal fmall moments or portions of time : he then 
^onfiders another line as generated by the unequal motion of a point, in 
fuch manner, that, in the aforefaid equal moments or portions of 
time, there maybe defcribed or cut off, from a given line, parts which 
fhall be continually in the fame proportion with the refpe£tive remain- 
ders, of that line, which had before been left : then are the feveral 
lengths of the firft line, the logarithms of the correfponding parts of 
the latter. Which defcription of them is fimilar to this, that the log- 
arithms are a feries of quantities or numbers in arithmetical progref- 
fion, adopted to another feries in geometrical progreffion. The firft 
or whole length of the line, 'which is diminiftied in goemetrical pro- ^ 
greiEon, he makes the radius of a circle, and its logarithm o or no- 
thing, reprcfenting the beginning of the firft or arithmetical line ; 
and the feveral proportiofjal -remainders of the geometrical line, are 
the natural fines of all t^e other parts of the quadrant decreafing down 
to nothings while the fucceffive increafing values of the arithmetical 
line, are the correfponding logarithms of thofe decreafing fines . fo that 
while (he natural fines decreafc from radius to nothing, their logarithms 
incrcafe from nothing to infinite. Napier made the logarithm of radius 
to be o, that he might fave the trouble of adding and fubtra£ting^ it 
in trigonometrical proportions, in which it fo frequently occurred ; 
' and he made the logarithms of the fines, from the intire quadrant down 
to o, to increafe, that fJ^cy might be pofitive, and fo in his opinion 
the cafier to manage, the fines being of more frequent ufe than the 
tangents and fecants, of which the whole of the latter and half the 
former would, in his way, be of a different afFedion from the fines ; 
for it is evident that the logarithms of all the fecants in the quadrant, 
and of all the tangents above 45*=^, or the half quadrant, would be ne- 
gative, being the logarithms of numbers greater than tihe radius, whofe 
logarithip is made equal to o or nothing. 

As to the contents of Napier's table ; it confifts of the natural fines . 
and their logarithms, fol: every minute of the quadrant. Like moft 
other tables, the arcs are coiitinued to 45 degrees from top to bottom 
on the left-hand fide of the pages, and then returned backwards from 
bottom to top on the right-hand fide of the pages : fo that the arcs and 
their complements, with the fines, natural and logarithmic, ftand on 
the fame line of the page, in fix columns; and in another column, in 
t'he^middle of the page, are placed the differences between the loga- 
rithmic fines and cofincs, on the fame lines, and in the adjacent co- 
lumns on the right and left; thus making in all feven columns in 
each page. Of thefe columns, the firft and feventh contain the arc 
and its complement, in degrees and minutes; the fecond and fixth, the' 
lUtural fine and cofine of each arc ; the third and fifth, the lagarichmig 

£ fine 



}6 HISTGRYOF 

fine and co/ine ; and the fourth, or middle column, the difierencc b?* 
tween the loguithmic line and cofinc which are in the third and iiftli 
columns. 

To elucidate the defcrjption, the fiift page of the table is here ii)- 
fertcd. 

Gr. o ■ + I — ■ 



Gr. 9 

Befidei the columns which are actually contained in this table, as 
above exhibited and defcribed, namely, the natural and logarithmic 
fines, and the diSercnccs pf thefc, the fame table is mailc to fcrve alfo 
for the logarithmic tang^ts and fccants of the whole quadrant, and 
- for the logarithms of common numbers. For, the fourth or middls 
column contains tl;e logarithmic tangents, being equal to the dif^ 
fences between the logarithmic fines and cofines when the )Dgzritiim 
pf radius is g> bccaufe cofi|ie: i^nc : ; radius ; t^geift, Uut is» 

m 



LOGARITHMS. aj 

t 

in logarithms, tangent r= fine — cofine. Alfo the logarithmic fines 
made negative become the logarithmic cofecants, and the logarithmic 
cofines made negativeare the logarithmic fecants^becaufe fine : radius : : 
radios : cofecant, and cofine : radius : : radius : fecant; that is, in 
logarithms, cofecant = o — fine r= — fine, and fecant = o — co- 
fine = — cofine. And to make it anfwer the purpofe o( a table of 
logarithms of common numbers, the author dire£ts to proceed thus :[ 
A number being given, find that number in any table of natural 
fines, oV tangents, or fecants, and note the degrees and' minutes in its 
arc J then in his table find the correfponding logarithmic fine, or tan- 
gent, or fecant, to the fame number of degrees and minutes ^ and it 
will be the required logarithm of the given number. 

Afiter his definitions and defcription of logarithms, Napier explains 
his table, and illuftrates the precepts with examples, (hewing how ta 
take out the logarithms of fines, tangents, fecants, and of common 
numbers; as alfo how to add and fubtra£l logarithms. He then pro- 
ceeds to teach the ufes of thofe numbers ; and firft, in finding any 
of the terms of three or four proportionals, (hewing how to multiply 
and divide, and to find powers and roots, by logarithms : 2dly, in 
trigonometry, both plane and fpherical, but efpecially the latter, in 
which he is very explicit, turning all the theorems fol* every cafe 
into logarithcns, computing examples to each in numbers, and then 
enumcfrating a fet of aftronomical problems of the fphcre which pro- 
perijr belong to each cafe. Napier here teaches alfo fome new theo- 
rems in fpherical trigonometry, particularly that the tangent of half 
the bale : tang, f fum legs : : tang. | dif. legs :.tang. | the alternate 
bafe; and the general theorem for what are called his five circular 
parts, by which he condenfes into one rule, in two parts, the theo- 
rehis' for all the cafes of right-angled fpherical triangles, which haci 
been feparately demonfl:rated by Pitifbus, Lanfbergius^ Copernicus^ 
ft^iomontanus, and others. 

Tbe^defeription and- ufc of Napier's canon being in the Latin lan- 
guage, they were tranflated into £ngli(h by Mr. Edward Wright, ^ 
an ingenious mathematician, and inventor of the principles of what 
has cbmmonly, though erroneoufly, been called Mercator's failing; 
He fent the tranflation to the author, at Edinburgh, to be revifed by 
him before pablication ; who, having carefully perufed it, returned 
it with with his app)*obation, and a few lines introduced befidesintd 
th^ tranflation. But, Mn Wright dying foon after he received it 
back^ it was after his death publiflied, together with the tables, but 
each number to one figure lefs, in the year 1616, accompanried with a: 
dedicatron, by bis fon Samuel Wright, to the £aff*-Indja Company^ 
as-alfd a- preface by Hefiry Briggs, of whom we (hall prefently have 
occ^fiicm to fipeik'more at large, on account of the great (hafe hrf 
lK>re in perfe<^ing ti^e logarithms. In this tranlTatlon Mr. Briggs gave 
alfo the defcription and draught of afcale that had been invented by Mr; 
Wright, and feveral otheif methods of his own, for finding the pro- 
portional parts to intermediate numbers, the logarithms having been 
onlypfinted ibr fuch numbers as were the natural: fines of each mi-* 

B 2- nutCv 



23 hist6ryof 

jnutc. And the note which baron Napier inferted in this Englifli edi-* 
tion, and which was not in the original, was as follows: ^^ But be* 
'* caule the addition and fubtraAion of thefe former numbers may 
<^ feem fomewhat painfull, I intend (\{ it (hall pleafe God) in a fecond 
*^ edition, to fet out fuch logarithms as fhall make thofe numbers 
'^ above written to fall upon decimal numbers, fuch as ioooocxxx>, 
*' 2000COOOO, 300CXXXXX), &c, which are eafie to be added or abated 
** to or from any other number." This note had reference to the 
alteration of the fcale of logarithms in fuch manner, that i fliould 
become the logarithm of the ratio of 10 to i, inftead of the number 
2 '3025 85 1 > which Napier had made that logarithm in his table, and 
which alteration had before been recommended to him by Briggs, a^ 
we (hall fee prefently. Napier alfo inferted a iimilar remark in his 
Rabdologla^ which he printed at Edinburgh in 1617. 

The following is the preface to Wright's book, which, as far as 
where it mentions the change from the Latin into Englifh, is a literal 
tranflation^f the preface-to Napier's original ; but what follows that, 
isadded by Napier himfelf. And 1 willingly infert it here, as it con- 
' tains a declaration of the motitres which led to this difcovery, and as 
the book itfclf is very fcsirce. *' Seeing there is nothing, (right well 
beloved ftudents in the mathematics) thai is fo troublcfome to Mathe- 
matical! pra6);ife, nor that doth more moleil and hinder Calculators, 
then the Multiplications, Divifions, fquare and cubical Extradions 
of great numbers, which befides the tedious expence of time, are, for 
the moft part fubjeft to many flippery errors. I began therefore to 
confider in my minde, by what certaine and ready Art I might remove 
thofe hindrances. And having thought upon many things to this pur- 
pofe, I found at length fome excellent briefe rules to be treated of 
(perhaps) hereafter. But amongft all, none more profitable then this, 
which together with the hard and tedious Multiplications, Divifions, 

and 



* Of this ingenious man I (hall here infert in a note the folJowing memoirs, as the/ 
have been tranilared from a Lntin piece taken out of the annals of Goimle and Cains 
College in Cambridge, viz. " liiis >ear (1615) died at London Edward Wright of Gar- 
veflon in Norfolk* formerly a fellow of iliis college; a man refpe^led by all for the inte- 
grity and timplicity of his manners, and alfo famoiis for his (kill in the mathematieal 
fcienccs : Jnfomuch that lie vva^ dcfcrvedly ftiled a moft excellent mathematician \fif 
Richard Hackluyt, the nvithor of nn original treatife of our £ngUlh narigations. What 
knowledge he had ncquired in tlie fclcnce of meclianicSi and how ufefutly he employed 
that knowledge to the public as well ns private advantage^ abundantly appear both from 
the writings he publifhed, and from the many mechanical o|)erations fliil extant^ wnich 
are fianding monuments of his great hiduOry and ingenuity. He was the firit undertaker 
of that difficult but ufeful work, by which a little river is brought from the town of Ware 
in a new Canal, to fupply the city of London with water; but by the tricks of others he 
was hindered froni. compleating the work he had begun. He was excellent both in coo* 
Crivance and execution, nor was he inferior to the moll ingenious mechanic in the making 
of inftniments, either of brafs or any ether matter. To his invention is owing whatever 
advantage Hondius*s geographical charts have above others; for it was our Wright thac 
taught JodocusHondius the method of conftru^ng them ^hich wut till then unknown^ 
but the ungrateful Hondius concealed the name of the true author, and arrogated th« 
glory of the invention to himfelf. Of this fraudulent practice the g(X)d man could not 
help complaining, and juiUy enough, in the preface to his treatife of the Corre^ion of 
Errors in the Art of Navigation ; which he compofed with excellent judgment, and after 
long experiencei to the great advancement of naval affairs. For the improvement of 

this 



LOOARI.T H Of S. M 

and Extractions ofrootes, doth alfo cafl afway from the yrorke !t felfe^ 
even the very numbers themielves that are to be multiplied, divided^ 
and refulved into rootes, and putteth other numbers in their place, 
which performe as much as they can do, onely by Addition and Sub- 
traction, Divifion by two, or Divifion by three; which fecret inven- 
tion, being (as all other good things are) fomuch the better as it fliall 
be the more common; I thought good heretofore to iet forth in La- 
tine for the pubiique ufe of Mathematicians. But now fome oSour 
Countrymen in this Ifland well affedted to thefe ftudies, and the more ^ 
pubiique good, procured a mod learned Mathematician to tranflate 
the fame into our vulgar Engliih tongue, who after he had finifhed 
it, fent the Coppy of it to me, to bee feene and confidered on by myfelf. 
I having moft willingly and gladly done the fame, finde it to bee moi): 
exa£l and precifely conformable to my minde and the original^ There* 
fore it may pleafe you who are inclined to thefe ftudies, to receive it 
from me and the Tranilator, with as much good will as we recom- 
mend it unto you. Fare yee welL** 

There are alfo extant copies of Wright's* tranflation with the date 
1618 in the title; but this is not properly a new edition, but only the 
old work with a new title page adapted to it (the old one being can- 
celled), together with the addition of (ixteen pages of new matter, 
called ^^ An appendix to the Logarithms, fhewing the practice of the 
calculation of triangles, and alfo a new and ready way for the exaCl 
findingoutof fuch linesand logarithmes as are not precifely to be found 
in the canons/' But we are not told by what author: probably it 
was by Briggs. 

Befides the trouble attending Napier's canon, in finding the propor- 
tional parts, when ufed as a table of the logarithms of common num- 
bers, and which was in part remedied by the fore-mentioned contri- 
vances of Wright and Briggs, it was alfo accompanied with another 
inconvenience, which arofe from the logarithms being fometimes + 



this art he ^2& appointed mathematical ledhirer by the Eaft India Compatiy, and read 
leAures in the houfe of that worthy knight Sir Thomas Smith, for which he had a yearif 
falarv of 50 pounds. This office he difcharged with great reputation, and mnch to the: 
fatisfa£lion of his hearers. He publifhed in Englifh a book on the doAriae of the fphere, 
«nd another concerning the conftru^on of fun-di^s. He ^fo prefixed an ingenious pre- 
face to the learned Gilbert's book on the load-ftone. By thefe and other hi» writings, he' 
bastranfmitted his fame to lateft pofterity- ' While he was yet a fellow of this college, he 
cpuld not be concealed in his private fludy, but was caUed forth to the public bufmefs of 
the kingdom, by the queen's majefly, about the year 1593* He was ordered to attend the 
carl of Cumberland in fome maritime expeditions. One of thefe he has given a faithful 
account of, in the way of a journal or epliemeris, to which he has pre/iXed. an elegant hy« 
drographical chart of his own contrivance. A little before his death he employed him- 
felf about an Englifh tranflation of the book of logarithms, then lately found out by the 
honourable baron Kapier, a Scotcliman, who had a great af&ftion for him. This ppilhu- 
mous wqrk of his was publilhed foon after, by his only fon Samuel Wright^ who was alfo a 
(cholarof this college. Hehad formed many other ufeful defigns, but was hindered by death 
from bringing them to perfe^ion. Of him it may be truly faid, that he "iludied more ta 
ferve the public than himfeif ; and though he was rich in fame, and in the promifes of tbcr 
great, yet he died poor, to the fcandal of an ungrateful age." 

Other anecdotes of him, as well as m^iny other mathematical authors, may be found 
in the curious hiilory of navigation by Dr. James Wilfony prefixed to Mr* Robertfon's 
otceUent treatifc on that fttbjo^ ... ' 

or 



1^ ^ HtStOHY OF 

9c addither, and fometinMS — er negativt, atul which r^uireJ ther€^ 
iore the knowledge of algebraic addition and fubtrsdion. And t\M9 
inconveni^mre was occasioned partly by making the logarithm of 
»dius to be o, and the' fines to decteafey and partly by the compen- 
dious manner in which^ the author had formed the table ; making the 
shree coluntne^of fines^ cofinea and tangents^ to ferve alfo for the other 
riiree of Oofe«ants, fecants, and cotangents* 

But this latter inconvenience was well remedied by John Speidell 
in his New Logariiims^ firfl: piibiiflied in 1619, which contained ali 
the fix columns, and in this order } fines, cofines^, tangents, cotan* 
gents, fecants, cofecants: and they werebefides made all pofitive, by bie* 
kig taken the arithmetical complements of Napier's, that is, they wore 
the remainders left by iubtrading each of thefe latter from looooooa. 
And the former inconvenience was more efledually removed by the 
ikid Speidell, in an additional table, given in the nxth impttsffioA of 
the former work-, in the year 1624. This was a table of Napier's 
logarithms for the round pr integer numbers r, 2, 3, }(, 5, &c, to looo, 
iegBther with their differences and arithmetical complements ; as alfo 
the halves of the faid logarithms, wit^ their differences and arithme- 
tical complements; which halves confequendy w^re t6e logarithms 
of the fquare roots of the faid numbers. Thefe logarithms are how«< 
ever a littlevaried in their form from Napier^s, namely, fo as to increafe 
Jir^m i» whofe logarithm is o^ inftead of decreafing fo i, or radius, 
w)iofe logarithm Napier made o likewife ; that is, Speidell's logarithm 
of any number ;r, is equal to Napier's logarithm of its reciprocal ^ : 
So that in this laft table of Speidell's, the logarithm of i being o, the 
logarithm of 10 is 2301584, the logarithm of 100 is twice as much or 
4605 169> and that of lOOO thrice as much or 6907753* 

This table is- now'commonly called hyperiolic logarithms^ becaufe 
tile numbers exprefs.the areas oetween theafymptote and curve of the 
Iiyperbola^ thojfe areas being limited by ordlnates parallel to the other 
afymptote, the^ordinates decreafing in goemetrical progreffion. But 
this is. an improper method of denominating them, as fuch areas may 
bjnadeto deooie any. fyAem of logarithms whatever, as we fhall fliew 
flK>re at large ia the proper' ptace^ 

In the year i6j[9^Rabert Napier, Ion of the inventor of logarithms, 
gubliihed ^ new^ edition of his late father^ Logarithmorum Canmis 
Jfeferifiis^ together with the promifed Logarithmotum Canonts Coft^ 
ftruStig^ and otoer mifcellaneous. pieces written by his father and Mr. 
£risg8.-<-Alfo one Bar thoiomew Vioceat, a bookfeller at Lugdunumt 
Off Lyons^ in France, primed' there an exa£t copy of the fame two 
worlus in one volumn, ia the year 16204 which was four years be-* 
fore the logarithms wei e carried to Fraoce by W ingate, who was there-* 
lore erroneeufiy faid to have firfl introduce^] them into that country. 
Butt I (hall treat more p;u:ticuUrly of the contents of this work afc^r 
rjiave enumerated, the other .writers on tiiis fort of logarithms. 

In 1618 or 1619, Benjamin Urfinus,- mathematician to the Eledoft 
of Brandenbu/gy publiibed, at Cologii, his Curfus MathematUus^ in 
which is contaiocd a copy of Napier's logarithms^ wich the. addition^ 

of 



L6 6AX.ITHMS. $t 

of fome libles of proportional parts. And in 1624 he prlkited at the 
fsLKkc place, his Trig^kprnetriaf with a table of natural fines and their 
logarithms, of the Napierian kind and form, to every ten feconds in 
the quadrant i which he had been at much pains in computing. 

In the fame year 1624, logarithms, of nearly the fame kind, were 
alfo publifhed, at Marpur^, by the famous John Kepler, mathemati- 
ci:|n to the Emperor Ferdmand the fecond, under the title of ChiUas 
Logarhbmorum ad T^tidtm Numeros Rotundas f prsemiffi Dtmonftrjitione 
kgitima Ortus LogaritbfMrum torumque Ufus^ f^c. and the vear follow* 
iny, a fupplement to the fame ; being applied to rouna or integer 
numbers, and to fuch natural fines as nearly coincide with them. 
Thefe are exadly the fame fort of logarithms as Napier's, being the 
fame logarithms of the natural fines of arcs beginning from the quad* 
rant, whofc fin^ or radius is looooooo, the logarithm of which is 
made o, and from thence the fines decreafing by equal differences, 
down to o, or the beginning of the quadrant, whilft their logarithms 
increase to infinity. So that the difference between this ubie and 
Napier's confifts only in this, namely, that in Napier's table the ^r^of 
the quadrant is dirided into equal parts, differing by one minute each, 
and confequevtly their fines, to which the logarithms are adapted^ are 
irrational or interminate numbers, and only exprefled by approximaite 
decimals; whereas in Kepler's table, the rmdims is divided into equal 
parts, which are confidered as perfe6l and terminate fines, having 
equsd differences, and to which terminate fines the logarithms are 
hefe adapted. By tht» means indeed the proportions for intermediate 
nun>bers and logarithms are eafier made, but then the correfponding 
arcs are not terminate^ but irrational, and only fet down to an ap* 
proximate degree. So that Kepler's table is more convenient as a 
table of the logarithms of ccunmon numbers^ and Napier's as the loga- 
rithmic fines of the arcs of the quadrant. In both tables the higarttofn 
of the ratio of 10 to i, t$ the fame quantity, samdy 23025852 s and 
as the radius, or greateft fine, is 1 0000000, whofe logarithm is made 
P9 the logarithms of the decuple parts of it will be found b]r adding 
23023852 continually, or multiplying this logarithm by 2, 3, 4r&c 
aiid hence the logarithm of i, the firft number, or fmaDeft fine, i^ tho 
table, is 161180959^ or 7 times 2302 &c* 

Befides the two columns, of the natural fines and their logarithms,, 
with the differences of the logarithms, this table of Kepler's confifts 
pjfp of three other columns.; the firft of which contains the neareft 
arcs, belonging to thofe fines, exprefied in degfeesy mimees anit- fe-» 
conds ; and the other two exprefs what parts of the radius eack fine 
is equal to, namely, the one of them in 241th parts of the radius, and. 
minutes and feconds of them ; and. the other in 60th parts of the nr^ 
dius, and minutes of tbem» As a fpecimen I have here extiaifted tfar 
}aft |]>9ge of the table^. printed exadly as in the work* 



AilCtTS' 



St 



HISTORY O^F 



80. 
80. 23 



lRCUS 

Circuli cum 
difierentiis. 



19. 
3' 

z o. 



•1 o. 



80. 44* 
2 1. 

81. 6. 



81. 99t 

81. S3- 

-25. 



82. i8. 

2 6. 

82* 45* 

•17. 



83- 13^ 

30. 

83- 43- 

3*' 



84* 16. 
36. 

84. S2> 

•41. 



4^. 



85. 

86. 22. 
|— r. 3. 

187. 26. 
»• 3 3- 
90. O* 



34 

^46 
I 2 

58 

5J 
33 

53 
*6 
6 

6 

54 
O 

2 O 

20 
40 

O 

30 

30 

9 

39 

54 

33 

4* 

IS 

45 
O 



TTrr 



vs 

feu Dumeri 
abfoiuti. 



98500.00 
98600.00 



Partes vice- 
iitne quartse. 



98700.00 
98800.00 



98900.00 

99000.00 



99100*00 
99200.00 



99300.00 

99400.00 



*3- 


38. 


24 


^3. 


39- 


50 


23- 


41. 


17 


^3- 


42. 


43 


23. 


44. 


10 


23. 


45- 


36 



23- 47- 
23- 48. 29 



99500.00 
99600.00 



99700.00 



99900.00 
loooqp.oo 



23. 


49. 


55 


23- 


5V 


22 


23. 


52. 


48 


23- 


54- 


14 



23- 5S- 41 



99800.0023. S7' 7 



23- 58. 34 
24. o. • 



LOGARITHMI 

cumdifierentiis 

101.58 

1511. Sen- 
ior. 47 
1409.89 + 

10 I. 37 

1308.52 + 

I 01.2 6 

1207.26 * 

TOI. 17 

1106.09 + 

. 10 I. 06 

1005.03 + 

z 00. 9 6 , 

904.0^7 + 
100. 85 

803.22 + 
100. 76 

702.46 
100. 65 

6oi.8i 

100. 56 

5Qi-25t 

100. 45 

4CX3.80 » 

■ 100. 35 

300-45 
100. t J 

20Q.2O 
I o o> 15 

100.05 
100. 05 

000000.00 



Paries 
fezagenarise 



59- 
59< 



6 
10 



59- 13 
59- »7 



59' 
59 



20 

24 



59' 
59 



28 
3« 



59. 


35 


59. 


38 


59. 


42 


59- 


46 


59- 


49 


59. 


53 


59. 


56 


60. 






To the table Kepler prefixes a pretty confiderable traft, contaming 
the conftrudion of the logarithms, and a demonftration of their pro- 
pterties and ftruSure, in which he confiders logarithms, in the true avi^ 
legitimate way, as the meafurcs of ratios, as ihall be £hewn more par- 
ticularly hereafter in the next part, where I (hall treat of the conltruc- 
tion of logarithms. 

Kepler alfo introduced the logarithmic calculus into his Rudolphinie 
tables, publifhed in 1627 > ^^^ inferted in that work feveral logarith- 
mic tables ; as firft, a table fimilar to that above defcribed, except 
that the fecond, or column of fines, or of abfolute numbers, is omitted, 
and inftead of it another column is added, (hewing what part of the 
quadrant each arc is equal to, namely the quotient, expreffed in in- 
tegers and fexagefimal parts, arifing from the dividing the whole 
quadrant by each given arc ; 2ci]y, Napier's table of logarithmic 
fines to every iminute of the quadrant ; alfo two other fmaller thbles 
adapted for the purpofes of eclipfes and the latitudes of the planets.-rr 
la this work alfo Kepler gives a fummary account of logarithms, with 

tb9 



LOGARITHMS. 33 

the defcriptioii and ttfe of thofe that are contained in thefe tables. 
And here it is that he mentions Juftus Byrgius, as having had loga- 
rithms before Napier publiihed them. 

fiefides the above^ fome few others publiflied logarithms of the fame 
krt about this time.— ^But let us now return to treat of the hiftory of 
the common or Brigg's logarithms, fo called becaufe he firft com« 
puted them, and firft mentioned them, and recommended them to 
Napier, inftead of the firft fort by him invented. 

Mr. Henry Briggs^ not lefa efteemed for his great probity and other 
eminent virtues, than for his excellent ikill in mathematics, was, at 
the time of the publication of Napier's logarithms, in 1614, profeiTor 
of geometry in Grefliam college in London, having been appointed 
the firft profeiTor after its inftitution ; which appointment he held till 
January liSao, when he was chofen, alfo the firft, Savilian profeiTor 
of geometry at Oxford, where he died January the 26th, 163J, aged 
about 74 years. 

On the publication of Napier's logarithms, Briggs immediately ap- 
plied himfelf tp the ftudy and improvement of them. In a letter to 
Mr. (afterwards archbifhop) Uiher, dated the loth of March 1615, 
he writes ** that he was wholly taken up and employed about the noble 
invention of logarithms lately difcovered,'* And again, *' Napier 
lord of Markinfton hath fet my head and hands at work with his new 
and admirable logarithms ; I hope to fee him this fummer, if it pleafe 
God ;- fbir I never faw a book which pleafed me better, and made me 
more wonder." Thus we find that Briggs began very earhy to com- 
pute logarithms : but thefe were not of the fame kind witfi Napier's, 
in which the logarithm of the ratio of lO to i was 2*3025851 &c. for 
in Brigg's firft attempt he made i the logarithm of that ratio ; and, 
from the evidence we have, he appears to be the firft perfon who 
ibrmed the Idea of this change in the fcale, which he prefently and 

{^eneroufly communicated both to the public in his ledlures, and to 
ord Napier himfelf, who afterwards faid that he alfo had thought of 
the fame thing ; as appears by the following extradb, tranflated fro^ 
the preface to Briggs's Arrthmetica Logarithmica ; ** Wonder not, fays 
he, . that thefe logarithms are different from thofe which the excellent 
baron of Marchtfton publiflied in his Admirable Canon. For when I 
explained the dcK^rine of them to my auditors at Grefliam College in 
London^ I remarked that it would be much more convenient, the lo« 
garithm of the fine total or radius being o (as in the Canon Mirtficus)^ 
if the logarithm of the 10th part of the faid radius, namely of ^^ 44^ 
21'', were 1 00000 &c. and concerning this I prefently wrote to the 
author ; alfo as foon as the ieafon of the year and my public teach-* 
ing would permit, I went to Edinburgh, where being kindly received 
by him, I ftaid a whole month. But when we began to converfe 
about the alteration of them, he faid that he had formerly thought of 
it and wiflied it \ but that he chofe to publifli thofe that were already 
done, till fuch time as his leifure and health would permit him to make 
others more convenient. And as to the nature of the change, he 
thought it more expedient that o fliould be made the. logarithm of i, 

F and 



3* 



HISTORY dt 



and looooo &c. the logarithm of radius, which I could not 6ut adCf-^ 
nowled^e was much better. Therefore, rejeAing thofe which I baJ 
before prepared, I proceeded at his exhortation to calculate tbefe; and 
the next fummer I went again to Edinburgh, to* (hew him the prfn- 
cipal of them : and fliould have been glad to do the fame the tfaini 
fummer, if it had pleafed God to fpare him fo long/' 

So that it is plain that Briggs was the inventor of the prefent feale 
of logarithms, in which i is the logarithm of the ratio of lo to i, 
and 2 that of I GO to i, &c. and that the ihare which Napier had inr 
them, was only advifing Brings fo begin al the leweft number i, and 
make the logarithms, or artificial numbers, as Napier had aHb called 
them; to tncreafe with the natural numbers inftead of decrenfingy 
which made no alteration in the figures that exprefled. Brrggs*s loga- 
rithms, but only in their afFedion or figns, changing them from ne« 
gative to poiitive; fo that Briggs's firft logarithms 
to the numbers in the fecond column of the annex- 
ed tablet, would have been as in the firft column; 
but after they were changed, as they are here m the 
third column; which is a change ot no efTential dif- 
ference, as the logarithm of the ratio of i& to i, the 
tadix of the natural fyftem of numbers, continues 
the fame, a change in the logarithm of that ratio be- 
ing the only circumftance that can elTentially alter 
the fyftem of logarithms, the logarithm of i being o. 
And the reafon why Briggs, after that interview, 
xejeAed what he had before done, and began anew^ 
was probably becaufe he had adapted his new loga- 
rithms to the approximate fmes of arcs^ inftead • of the round or in- 
teger numbers, and not from their being logarithms of another fyftem^ 
as were thofe of Napier. 

On Briggs's return from Edinburgh to London the fecond time, 
namely in 1617, he printed the firft ihouiand logarithms, to eighe 
places of figures befides the index, under the title of Lpgaritbmorum 
Chilias prima. But thefe feem not to have been publifhed till aftet 
the death of Napier, which happened on the third of April 1618, as 
before-faid ; for in the preface to them Briggs fays, *^ why thefe loga- 
rithms difter from thofe let forth by their moft illuftrious inventor, of 
ever refpeflful memory, in his Canon Mirificm^ it is to be hoped 
his pofthumous work will fhortly makeappear»" And as Napier, af- 
ter communication had with Briggs on the fubjed^ of altericfg the 
fcale of logarithms, had given notice, both in Wright's tranflation, 
and in his own Rabdologia^ printed in 16 17, of his intention to alter 
the fcalc (though it appears very planely that he never intended to 
compute any more), without making any mention of the ihare which 
Briggs had in the alteration, this gentleman modeftly gave the above 
hint. But not finding any regard paid to it in the faid pofthumous 
work, publifhed by lord Napier's, Ion in 1619^ where the alteration. is 
again adverted to, but fKli without any mention of Briggs; this gen- 
tleman thought he could not do lefs than ftate the grounds of that al- 
teration 



B 


NlHD. 


N" 


n 


to« 


— « 


3 


•001 


-3 


2 


•01 


-2 


I 


•I 


— I 





I 





— I 


zo 


I 


— 2 


100 


2 


-3 


1000 


3 


•^« 


10" 


n 



LOGARITHM p. j^ 

teratioii liimfelf, as they are alcove extraded from bis work publifhed 
in 1624. 

Thus, upon the whole matter, it feems evident that Mr. Briggs^ 
whether he had thought of this improvement in the eonftrudlion of 
logarithms, of making i the logarithm of the ratio of 10 to i, beforo 
lord Mapier, or not, (which is a fecret that could be known only to 
Napier himfelf), was the firft perfon who communicated the idea of 
£ich an improvement to the world; and that he did this in his lec- 
tures to bis auditors at Grefham College in the year 161 5, very foon 
after his perufai of Napier's Canon Mirificus Logaritbmorum in the 
year 1614. He alfo mentioned it to Napier, both by letter in the 
fame year, and on his firft vifit to him in Scotland in the fummer of 
the year 161^, when Napier approved the idea, and faid it had already 
occured to himfelf, and that he had determined to adopt it. It would 
therefore have been move candid in lord Napier to have told the world 
in the fecond edition of this book, that Mr. Briggs had mentioned 
this improvement to him, and that he bad thereby beep confirmed 
in the refolution he had already taken, before Mr. Briggs's commu- 
nication with him, to adopt it in that his fecond jedition, as being 
better fitted to the decimal notation of arithmetic whiqh was in gene- 
ral ufe. Su^h a declaration would have been but a piece of juilice 
<o Mr. Briggs; and the not having made it, cannot but incline us to 
fufpe^ that lord Napier was defirous that the world fhpuld afcribe to 
him alone the merit of this very ufeful improvement of the loga- 
rithms, as well as that of having originally invented them; though^ 
if the having firft. communicated an invention to the world be fuflici* 
ent to intitle a man to the honour of having iirfl invented it, Mr. 
3dggs had the better title to be called the firft inventor of this happy 
improvetnent of logarithms. 

In 1620, two years after the Chi lias Prima of Briggs came out, 
Mr. Edward Gunter publifhed his Canon of Triangles, which contains 
the artificial or logarlthmetic fines and tangents, for every minute, to 
ibven places of figures, befidesthe index, the logarithm of radius bet- 
ing 10*0 &Ct Thefe logarithms are of the kind laft agreed upon by 
Napier and Briggs, and they were the firft tables of logarithmic fines 
and tangents that were publifhed of this fort. Gunter alfo in 1623 
reprinted the fame in his book D^ SeSlore it Rqdioy together with the 
CA/VwPriwfl of his old colleague Mr. Briggs* he being profefTor of 
Aftronomy at Grefham college when Briggs was profeflbr of Geo- 
metry there, Gunter having been elefted to that office the fucth of 
March 1619, and enjoyed it till his death, which happened on the 
tenth of December 1626, about the forty-fifth year of his age. In 
1623 alfo Gunter applied thefe logarithms of numbers, fines and tan- 
gents, to ftraight lines drawn on a ruler; with which proportions in 
common numbers and trigonometry were refolved by the mere appli- 
cation of a pair of compafTes; a method founded on this property, that 
the logarithms of the terms of equal ratios are equidifferent. This 
inftrument, in the form of a two-foot fcale, is now in common ufe 
for navigation and other purpofes, and is commonly called the Gun- 



$6 HISTORYOP 

tcr. He alfo greatly improved the k&or for the fame ufes. Gunter 
was the firft who ufed the word co-Jine for the fine of the complement 
of an arc. He alfo introduced the ufe of arithmetical complements 
into the logarithmical arithmetic, as is witneflfed by Briggs, chap. xv. 
Arith. Log. And it has been faid that he ftarted the idea of the 
logarithmic curve, which was fo called becaufe the fegments of its 
axis are the logarithms of the correfponding ordinates. 

The logarithmic lines were after ways drawn in various other ways* 
In 1627 they were drawn by Wingate, on two feparate rulers. Aiding 
dgainft each other, to fave the ufe of compafles in refolving proper* 
tions. They were alfo in 1627 -applied to concentric circles^ by 
Oughtred. Then !n a fpecial form by a Mr. Milburne of York-? 
{hire, about the year 1650. And laftly, in 1657, on the prefent flid-r 
Ingrule, by Seth Partridge. 

The difcoveries relating to logarithms were carried to France by 
Mr. Edmund Wingate, but not firft of all as he erroneoufly fays in 
the preface to his book. He publi(hed at Paris, in 1624) two fmall 
traAs in the French language; and afterwards at London, in 1626^ 
an EngHih edition of the fame, with improvements. In the firft of 
thefe he teaches the ufe of Gunter's ruler; and in the other, that of 
Briggs's logarithms, and the artificial fines and tangents. Here are 
contained alfp tables'of thofe logarithms, fines, and tangents, copied 
from Gunter. The edition of thefe logarithms printed at London 
in 1635, and the former editions alfo I fuppofe, has the units figures 
difpofed along the tops of the colunins, and the tens down the roar- 
gins, like our tables at prefent; but the whole logarithm^ which wa$ 
only to (ix places of figures, in the angle of meeting. Which is tbq 
£rft inftance that I have feen of this mode of arrangement. 

fiut proceed we now to the larger ftruSureof logarithms. 

Briggs had continued from the beginning to labour, with great in* 
duftry, at the computation of thofe logarithms of which he before 
publifhed a fhort fpecimen in fmall numbers. And in 1624 ^^ P^^* 
duced his Ariihmetica Logaritbmicaj a ftupendous work for fo fhort a 
time! containing the logarithms of 30000 natural numbers, to four- 
teen places of figures befides the index, namely from i to 20000, and 
from 90000 to 100000 ; together with the differences of the loga« 
rithm^. Some writers fay that there was another chiliad^ namely from 
looooo to loiooo; but none of the copies that I have feen have morQ 
than the 30000 above-mentioned, and they were all regularly tcr*- 
minated in the ufual way with the word finis. The preface to thefo 
logarithms contains, among other things, an account of the altpratioi| 
made in the fcale by Napier and himfelf, from which we before gav^ 
an extrad ; and an earneft foil citation to others to undertake th^ com- 
putation for the intermediate numbers, ofiering to give inftrudions, 
and paper ready ruled for that purpofe, to anjr perfons fo inclined tq 
contribute to the completion of fo valuable a work. In the introduc<r 
tion he gives alfo an ample treatife on the conftru&ion and ufesof th^fe 
logarithms, which will be particularly defcribed hereafter* " By 

this ihvitatiop^ ai>4 other means, be ha() hopes of ^o|le£iing ma^erial^ 



tOOARITHMSt jy 

for tilt logarithms of the intermediate 70000 numbers, whilft he (hould 
employ his own labour more immediately upon the canon of logarith-* 
snic fines arid tai\gents, and fo carry on both works at once; as indeed 
they were both equally ne^efTary, and he himfelf was now pretty far ' 
advanced in years. 

Soon after this Adrian Vlacq, or Flacky of Gouda in Holland, corner 
pleated the intermediate feventy chiliads, and republifhed the Jritbrm^ 
iica LQgarithmica at that place, in 1627 and 1628, with thofe inter- 
mediate numbers, making in the whole the logarithms of all numbers 
to loooco, but only to ten places of figures. To thefe was added a 
tabje of artificial fines, tangents and fegants, to every ininute of th^ 
quadrant* 

Briggs himfelf lived alfo to compleat a table of logarithmic (inet 
^nd tangents for the hundredth part of every degree, to fourteen places 
of figures beildes the index ; together with a table of natural fines for 
the fame parts to fifteen places, and the tangents and fecants for the 
fame to ten places; with the conftru£lion of the whole, Thefe tables 
were printed at Gouda^ under the care of Adrian Vlacq, and moftly 
^nifhed ofFbefore 1631^ though not publifbed till 1633. But his death, 
which then happened, prevented him fromcompleating the application 
and ufesof them. However, the performing of thi^ office, when dy-' 
ing, he recommended to his friend Henry Gellibrand, who was then . 
profefibr of Aflronomy in Grefham college, having fucceeded Mr* 
Gunter in that appointment. Gellibrand accordingly added a pre* 
face and the application of the logarithms to plane and fpherical tri« 
gonometry &c. and the whole was printed at Gouda, by the fame 
printer, and brought out in the fame year, 1633, as the Tngononutri^ 
^rtificialis of Vlacq, who had the care of the prefs, as abovefaid* 
This work was called Trigon^nutria Brit^nnica ; and befides the arcs 
in degrees and centefms of degrees, it has another column containing 
the minutes and feconds anfwerine to the feveral centefms in the firft 
polumn. 

In 16339 as mentioned above, Vlacq printed, at Gouda in Hol- 
land, his Trigonometria Artificialis : five Magnus Canon Trianguhrum 
{sogaritbmitus ad Decadas Secundorum Scrupulorum conJtruQus. This 
j/iOxW contains the logarithmic fines and tangents to 10 places of fi* 
gures, with their dinerences, for every 10 feconds in the quadrant* 
To them is alfq added Brigg's table of the firfl 20000 logarithms, but 
carried only to 10 places of figures befides the index, with their dif- 
ferences. The whole is preceded by a defcription of the tables, and 
fhe application of them to pbne and fpherical trigonometry, chiefly 
extraded from Briggs's Trigonometria Britannica^ mentioned above. 

Gellibrand publimed alfo, in iS^Sj -^« Inftitution Trigonometricall^ 
containing the logarithms of the firn loooo numbers, with the natural 
ifines, tangents and fecants, and the logarithmic fines and tangents» 
for degrees and miniites, all to feven places of figures befides the 
index I as alfo other tables proper for navigation ; with the ufes of the 
91^ hole. Gellibrand died the 9th of Februar]^ 1636, in the 40th year 
of bis age,^ \q the gfeat lofs of the mathenuti^al world. 

Befides 



j8 HISTOH-YOf 

Befid^i^the ptrfoDS hitherto mentioned, who were moftly computeri 
of logarithms, many others have alfo publiflied tables of thofe artifi« 
cial numbers, more or lefs compleat, and fom«times improved and 
varied in. the manner and form of chenu ^ I fhall here juft advert to a 
few of the principal. 

In 1626, D. Henrion publiihed, at Paris, a treatife concerning 
Priggs's logarithms of common numbers from i to 20000, to eleven 
places of figures ; with the fi|ies and tangents to eight places only. 

In 1 631 was printed at London, by one George Miller, a book, 
containing Brigg's logarithms, with their differences, to ten places of 
figures befides the index, fbr 4II numbers to ipopoo ; as alfo the logar« 
ithmic fines, tangents and fecants for every minute of the quadrant | 
Virith the explanation and ufes in EngliQi. 

The fanie year 1631, Richard Norwood publiihed his Trigommetriei 
in which we f^nd Briggs's logarithms for all numbers to 10000, and 
for the fines, tangents and fecants to every minute, both to fevei) 
places befides the index.-— In the conclufion of the trigonometry he 
complains of the unfair practices of printing VUcq's hook in 1627 ^' 
1628, and the book mentioned in the lafl article. His words are, 
*^ Now whereas 1 have-he^e, and in fundry places in this book, cited 
Mr. Briggs his Arithmitica Logarithmicay (left I may feem to abuie the 
reader) you are to underfland n9t the book put forth aboqt a month 
fince in Englifh, as a tranflation of his, and with the fame title ; be« 
ing nothing li|(;e his, nor worthy his name ; but the book which him-* 
felf put forth with this title in Latin, being printed at London, anno 
1624* And here I have jufl occafion to blame the ill dealing of thefe 
men, both in the matter before mentioned, and in printing a fecon4 
edition of his ArithmeticaLogarithmica inXatin, whilflhe lived, againfl 
his mind and liking ; and brought them over to fell, when the fith were 
unfbld ; fo fruftrating thofe additions which Mr. Briggs intended iq 
his fecond edition, and moreover leaving out fome things that were in 
the firft edition of fpecial moment, A pra£lice of very ijl confequence, 
and tending to the great diiparagement of fuch as take pains in thi^ 
Jiind.** 

Francis Bona venture Cavalerius publifhed at Bologna, in 1632, 
his Dinlforium GimraU Uranometricum^ in which are tables of Brigg's 
logarithms of fines, tangents, fecants, and verfed fines, each to eight 
places, for every fecond of the firfl five minutes, for every five fecond$ 
from five to ten minutes, for every ten feconds from ten to twenty 
minutes, for xvery twenty feconds from twenty to thirty minutes, for 
every thirty ieconds from 30' to i® 30', and for every minute in the 
refl of the quadrant; which is the firft table of logarithmic verfed 
fines that I know of. In this book are contained alio the logarithms 
of the firfl ten chiliads of natural numbers, namely from i to 1000O9 
difpofed in this manner, all the twenties at top, and from i to 19 on 
the fide, the logarithm of the fum being in the fquare of meeting. 
In this work alfo I think Cavalerius firfl gave the niethod of finding 
the area or fpherical furface ^onCftined by various arcs defcrib^d on 
tl^e fufface of a ipbe^e* 

Alfo 



LOOARlTliMS^ 



39 



Alfo in the Trig9k9$iufria of the fame author, printed in 164^, be-' 
fides the logarithms of numbers from i to looo, to eight places, with 
their differences, we find both natural and logarithmic fines, tan-s 
gents and fecants, the former to feven and the latter to eight places t 
liamely, to every 16" of the firft 30 minutes, to every 30^' from yr 
to i^ } and the fame for their complements, or backwards through 
the laft degree of the quadrant ; the intermediate SB9 being to every 
tninute only* 

Mr. Natnaniel Roe, << Paftor of Benacre in Suffolke/^ alfo redu-* 
ced the logarithmic tables to a contraded form, in his TabuUe L9gar* 
ithmUitj printed at London in 1633. Here we have Briggs's logar- 
ithms of numbers from i to lOooOO, to eight places j the fifties' 
placed at top, and from i to 50 on the fide ; alfo the firft four figures 
of the lo^rithms at top, and the other four down the columns. Tfaejr 
contain alfo the logarithmic fines and tangents to every iCOth part of 
degrees, to ten places. 

Ludovicus Frobenius publiflied at Hamburgh, in 1634, his Clavh 
t/mverfa Trigonometrimy containing tables of Bri^s's logarithms of 
numbers from i to 2000; and of fines^ tangents, and fecants, for 
every minute ; both to feven places. 

But the tables of logarithms of common numbers was reduced to 
its moft convenient form by John Newton, in his Trigonomttria Brl» 
taitntcai printed at London in 1658, having availed himfelf of both 
the improvements of Wingate and Roe, namely, uniting Wingate's 
difpofition of the natural numbers with Roe's contraAed arrangement 
of the logarithms, the numbers being all difpofed as in our beit tables 
at prefent, namely, the units along the top of the page, and the tens 
down the left-hand fide, alfo the firft three figures of each logarithm 
in the firft column, and the remaining five figures in the other co- 
lumns, the logarithms being to eight places. This work contains 
alfo the logarithmic fines and tangents, to eight figures befides the 
index, for every :100th part of a degree, with their differences, and 
for 1000th piirts in the firft three degrees. — In the preface to this work, 
Newton takes occafion, as Wingate and Norwood had done before^ 
as well as Briggshimfdf, to cenfure the unfair pradices of fome other 
publifhers of logarithms. He fays, *^ In the fecond part of this infti- 
tution, thou art prefented with Mr. Gellibrand's Trigonometric, 
faithfully tranflated from the Latin copy, that which the author him- 
ielf publiftied under the title of TrigMametna Sritannicay and not that 
which Vlacq the Dutchman ftiles Vtrigonomitria jtrtlficialisy from 
whofe corrupt and imperfed copy that feems to be tranflated, which is 
amongft us generally known by the name of GeUihrands Trigonomitrf, 
but thofe who either knew him, or have perufed his writings, can tef- 
tifie that he was no admirer of the old fexagenary way of working, nay, 
that he did preferre the decimal way before it, as he hath abundantly 
teftified in all the examples of this his Trigonometry, which differs 
from that other which Vlacq hath publiilv^d, and that which hath 
hitherto borne his name in Englifh, as in the form ; fo likewife in 
the matter of it i for in the twolaft-mentioned editions, there is fome- 

thing 



i|o ISISTORV OF 

thing lef]^ out in the fecond chapter of plain triangles^ (he (h\t3 chip^ 
ter wholly omitted, and a part of the third in the ipherical, but in tbisT 
edition nothing, fomething we have added to both^ by way of expla- 
nation and demonftration." 

In 1670, John Caramuel publifiied his Afathe/ts Nova^ in which are 
contained loqo logarithms both of Napier's andBriggs's form, as alfo 
1000 of what he calls the Perfe£t logarithms, namely the fame as 
thofe which Briggs iirft thought of, which differ from the laft only^ia 
this, that the one increaies while the other decreafes, the radix^ or lo« 
garithm of the ratio of 19 to i, being the fame in both* / 

The books of logarithms have fince become very numerous, but the. 
logarithms are m<mly of that ibrt invented by Briggs, and which aref 
now in common ufe. Of thefe the moft noted for their accuracy or 
vfefiilnefs, befides the works above-mentioned, are Vlacq's fniall vo- 
lume of tables, particularly that edition printed at Lyons in 1670 ; 
alfo tables printed at the iame place in 1760; but mod efpecially the 
tables of Sherwin and Gardiner* Of thefe, Sherwin's Matbematicat 
tables in 8vo^ form the moftcompleat colledlion of any, containing^ 
befides the logarithms of all numbers to loiopo, the fines, tangents^ 
fecants,'and verled fines, both natural and logarithmic, to every mi« 
Bute of the quadrant. The firft edition was in 1706 ; but the third 
edition, in 1742, which wasrevifed by Gardiner, isefteemed themoft 
corred of any : as to the laft or fifth edition, in 1771, it is fo erro^ 
neoufly printed, that no dependence can be placed in it, and it is the 
moft inaccurate book of tables I ever knew ; I have a lift of feveral 
thoufand errors which I have corrected in it. 

Gardiner alio printed at London, in 1742, a quarto volume of 
^^ Tables of Logarithms, for all numbers from i to 102100, and for 
tiie fines and tangents to every ten feconds of each degree in the quad* 
lant ;.as alfo, for the fines of the firft 72 minutes to every fingle fe* 
cond : with other ufeful and necefifary tables ;'' namely> a table of 
LogifticftI Logarithms, and three fmaller tables to be ufed for find- 
ing the logarithms of numbers to twenty places of figures* Of thefe 
tables of Gardiner, only a (mall number was printed, and that by 
fubfcription ; and they are now in the higheft eftimation of any loga- 
rithms for their accuracy and ufefulnefs. 

An edition of Gardiner's colledion was alfo elegantly printed aC 
Avignon in France, in 1770, with fome additions, namely, the fine& 
and tangents for every fingle fecond in the firft four degrees, and a 
fmall table of hyperbolic logarithms copied from a treatife on Flux- 
ions by the late ingenious Mr. Thomas Simpfon : but this is not quite 
fo correal as Gardiner's own edition* The tables in all' thefe books 
are to feven places of figures^ 

^' The logarithmic canon ferves to find readily the logarithm of 
any affigned number ; and we are told by Dr. Wallis, in die fecond 
▼olume of his Mathematical Works, that an antilogarithmic canon, 
or one to find as readily the number ^orrefponding to every loearithm, 
was begun he thinks by Mr. Harriot the algebraift (who died in 162 1 ) 

' completed by Mr. Walter Warner, tU editor of Harriot's works, 

before 



LOGARITHMS. 41 



before 1640 ; which ingenious performance it feems was loft, for want 
of encouragement to publifh it." 

<* A fmail fpecimen.of fuch numbers was publlfbed in the Philofo* 
phical Tranfa6tions, for the year 1714.) by Mr. Long of Oxford i but 
it was not till 1742 that a complete antilogarithmic canon was pub- 
lifhed, by Mr. James Dodfon, wherein be has computed the numbers 
correfponding-to every logarithm from i to locxxx), to ii places of 
figures." 

Since the preceding account was written/and wbilft it wa^ in the 
prefs, there has been printed at Paris, ^* Tables Portatives de Logar- 
ithmes, publiees a Londres par Gardiner," &c. This work is moft 
beautifully printed in a neat portable 8vo volume, and contains all 
the tables in Gardiner's 4to. volume. With ibme additions and improve- 
ments. But with what degree of accuracy remains yet to1>e deter- 
mined. And on this, as well as feveral other occaiions, it is but juf- 
tice CO remark the extraordinary fpirit and elegance with which the 
learned men and the artifans of the French nation undertake and exe- 
cute works of merit. 



THE CONSTRUCTION 

O F 

LOGARITHMS, &c. 

HAVING defcribed the feveral forts of logarithms, their rife and 
iuvention, theirnature and properties, and given fome account 
of the principal early cultivators of them, witli the chief collections 
that have been publilhed of fuch tables; I proceed now to deliver a more 
particular account of the ideas and methods employed by each author^ 
;ind the peculiar modes of conftru£tion which they made ufe of. 
And nrfl of the great inventor himfelf, lord Napier. 

Napier^ r Cofi/iru^ion of Logarithms. 

The inventor of logarithms did not adapt them to the feries of na- 
tural numbers i» 2, ^ 4, 5,' &c, as it was not his principal idea to 
extend them to all arithmetical operations whatever ; but he confined 
his labours to that circumftance which firft fuggefted the neccffity of 
the invention, and adapted his logarithms to the approximate numoers 
expreiiing the natural fines of every minute in the quadrant, as they 
had been fet down by former writers on trigonometiy. 

The fame reftri£ted idea was purfued through iiis method of con* 
ilruding tlie logarithms. As the lines of the fines, of all'arcs, are parts 
.of the radius, or fine of the quadrant, therefore called the Jinus totus or 
whole fine, he conceived the line of the radius to be defcribed, orrufi 
over, by a point moving along it in fuch manner, that in equal portions 
of time it generated, or cut off, parts in a decreafing geometrical pro- 
greffion, leaving the feveral remainders, or fines, in geometrical pro- 
greflion alfo; whilft another point, in an indefinite line, defcribed equal 
parts of /> in the fame equal portions of time ; fo that the refpedive fums 
of thefe, or the whole line gentrated,. were always the arithoieticals or 

G logarithms 



4* 



CON S TRl/CTION' OF' 



5 
6 

7 



z 



fcc 



logarithms of thofc fines. Thus, az is the given radius tfpon Sines, ttf^^ 
which all the fines are to be taken, and A&fc the indefinite 
line containing the logarithms ; thefe lines being each gene- 
rated by the motion of a pointy beginning at A, a. Now at 
the end of the ift, 2d, 3d, &c, moments, or equal fttiall 
portions of time, let the moving points be found at the 
places marked i, 2, 3, &c ; then za, zi, z2, 23, &c, will be 
the fcries of natural fines, and Ao (or o), Ai, A2, A3, &c, 
will be their logarithms ; fuppofing ^bc point which gene- 
rates az to move every wliere with a velocity decreafing in 
proportion to its dfiftance from z, namely, its velocity in the 
points o, I, 2, 3, &c, to be refpedively as the diftances zo, 
21, z2, Z3, &c, whilft the velocity of the paint generating 
the logarithmic line A&c, remain? condbuitly die fame as at 
firft in the point A or o. 

Hitherto the author had not fully Ihnired his fyftem or Icale of 
logarithms, having only fuppofed one condition or bmitation, namely^ 
that the logarithm of the radius az Ihould be o. But two independant 
conditions, no matter what they arc,' were neceflary to limit the fcalc 
or fyftem of logarithms . It did not ocpur to him, that it was proper 
to form the other limit by affixing fomc particular value to an afiigned 
number, or part of the radius : but as another condition was neccf-' 
. i'ary, be affumed this for it, namely, that the two generating points 
fliould begin to move at a, A with equal velocities ; or that the incre- 
ments ai, Ai, defcribed in the firft moments, ihould be equal ; as he 
thought this circumftance would be attended with fome little eafe in 
the computation. And this is the reafon that, in bistable, the natural 
fines and their logarithms, at the compleat quadrant, have equal dif- 
ferences ; and this is alfo the reafon why his fcale of logarithms happens- 
accidentally to agree with what have fince been called the hyperbolib 
logarithms, which have numerical differences equal to thofc of tbeJr 
natural numbers at the beginning. ; except only that thefe latter increafe 
with the natural numbers, and his on the contrary decreafe ; the lo- 
garithms of the ratio of 10 to i being the fi^mc in both, namefy 
2-30258509. 

And here by the wiy it may be obferved, tliat Napier's mannet of 
conceiving the generation of the lines of tlte natuial numbers and 
their logarithms, by the motiok of points, is very fimilar to the man- 
ner in which Newton afterwards confidered the generation of magjnl- 
tudes in his doftrine of fluxions ; and it is alfo remarkable that, in 
art. 2. of the Habitudines Logartthmorum W fuorum naturaliumnunuro* 
rum invicem in the appendix to the Con/Iru^io Logartthmorum^ Napier 
fpeaks of the velocities of the increments or decrements of the loga- 
rithms, in the fame way as Newton docs, namely of his fluxions, 
where he fliews that thofc velocities, or fluxions, are invcrfely as 
the fines or natural numbers of the logarithms ; which is a neceiiary 
confequence of the nature of the generation of thofc lines as defcribed 
above ; with this alteration however, tliat now tlie radius az muft be 
confidered as generated by an equable motion of the point, and the 
indefinijte line A &c by a motion increafing in the fame ratio as the 

other 



LOGARITHMS. 



m 



ether before decreafed ; which is a fuppofition that Napier mufl have 
had in view when he dated that relation of the fluxions. 

Having thus limited his fyftem, Napier proceeds, in the pofthumous 
work of 16199 to explain his conftruAion of the logarithmic canon ; 
and this he eife&s in various ways, but chiefly by generating, in a 
very eafy manner, ^ feries of proportional numbers and their arithme* 
ticals, or logarithms ; and then finding, by proportion, the logaritlims 
to the natural lines, from thofe of the neareft numbers among the 
original proportionals* 

After defcribing the necei&ry cautions he made ufe of to preferve a 
fufhcient degree of accuracy, in fo long and complex a procefs of calcu- 
jiation; fut:h as annexing feveral ciphers, as decimals feparated by a point 
^o his primitive numbers, and rejefling the decimals thence refulting 
after tlie operations were compleated, fetting tbe numbers down to the 
neareft unit in the laft figure ; and teachiag the arithmetical procefTes 
of adding, fubtraAing, multiplying, and dividing the limits hetween 
i^hich certain unknown numbers muft lie, fo as to obtain the limits 
between which tlie refults muft alfo fall ; I fay, after defcribing fuch 
particulars, in order to clear and fmootli die way, he enters on the great 
iield of calculation itfelf. Beginning at radius icxxxx>oo,i he firft con- 
ftruAs feveral defcending geometrical feries, but of fuch a nature that 
the^ are all quickly ToriQed by an eafy continual fubtra£tion, and a 
diviiion by 2, or by 10, or 100, &c, which is done by only removing' 
the decimal point fo i^any places towards the left hand, as there are 
ciphers in the divifor. He conftru£ls three tables of fuch feries : The 
iirit of thefe confifts of 100 numbers, in the proportion of radius'to 
radius minus 1, or of looooooo to 9999999 ; all which are found by 
only fubtrading from each its icxxxxxxDth part, which part is alfo 
found by only removing each figure 7 places lower : the laft of thefe 
100 proportionals is found to be 9999900*0004950. 

The 2nd table contains 50 



No. 



3 
4 

50 

100 



Firft Table. 

1 0000000.0000000 

9999999.000000C 

9999998. OOOOOO I 

9999997.0000003 

&(; till the looth 

term, which will bt 

9999900.00049 CO 



2d Table, 

I coooooo.oooooo 

9999900*000000 

9999800*001000 

9999700*003000 

&c to the 50th term 
9995001.222927 



numbers, which are in the 
continual proportion of the 
firft to the laft in thefirft table, 
namely, of 10000000.0000000 
to 9999900.0004950, or near- 
ly the proportion of 100600 
to 99999 ; thefe therefore are 
found by only removing the figures of each number 5 places lower, 
and fubtrafting them from the lame number : the laft of thefe he finds 
to be 9995001.222927. And a fpecimen of thefe two tables, is here 
annexed. 

The 3d table confifts of 69 columns, and each column of twenty- 
cne numbers or terms, which terms, in every column, are in the con- 
tinual proportion of 1 0000 to 9995, that is, nearly as the firft is to 
the lafi in the 2d table; and as loooo exceeds 9995 by the 2000th 
part, the terms in every column will be conftruAed by dividing each 
upper number by 2, removing the figures of the quotient 3 places 
lower, and then fubtrafting them ; and in this way it is proper to con- 
ftruft only the firft column of 21 numbers, tlie iaft of which will be 

G z - . 9900473-5780; 



44 



CONSTRUCTION OF 



9900473.5780: but the ift, 2d, 3d, &c, numbers in all the colamtis, 
arc in the continual proportion of 100 to 99, or nearly the proportion 
of the firil to the lau in the iirft column ; and therefore thefis will be 
found by removing tlie figures of each preceding number 2 places 
lower, and fubtradting them, for the like number in the next column. 
A fpecimen of this 3d table is as here below. 



The 3d Tabic. 



Terms 



I 

z 

3 

4 

5 
&c 

£1 



I ft Column. 



10000000.0000 
9995000.0000 
9990002.5000 
9985007.4987 
9980014.9950 

kc till 
9900473.5780 



2nd Column. 



9900000.0000 
9895050.0000 
98901024750 
98851574237 
98802 14.845 1 

&e 
9801468.8423 



3d Column. 



9801000.0000 
9796099.5000 
97912014503 
9786305.8495 
9781412.6967 

&c 
9703454.1539 



3cc till the 69 Column. 



fcc for the 
4th, 5th, 6c hy 
7th, &c col. 

till the lad 



or 



1^ 



5048858.8900 
5046334.4605 
504381 1.2932 
5041289.3879 

5038768.743s 
ice 

4998609.4034 



Thus he .had, in this 3d table, interpofed between the radius and 
its half, 68 numbers in the continual proportion of 100 to 99 ; and 
interpofed between every two of thefc 20 numbers in the proportion 
of loooo to 9995 : and again, in the 2d table, between looooooo and 
9995000, the two firft of the 3d table, he had 50 numbers in the pro- 
portion of 1 00000 to 99999 : And laftly, in the ift table, between 
1 0000000 and 9999900, or the 2 firft of the 2d table, 100 numbers 
in the proportion of 1 0000000 to 9999999. That is, in all about 
1600 proportionals ; all found in the moft fimplc manner by little 
more thaneafy fubtradions; which proportionals nearly coincide with 
all the natural fines from 90' down to 30^ 

To obtain the logarithms of all thofe proportionals, he demonftrates 
feveral properties and relations of thp numbers and logarithms, and 
iliufirates the manner of applying them. The principal of thefe pro^ 
perties are as follows : ift, that the logarithm of any fine is greater 
than the difference between that fine and the radius, but lefs than the 
faid difference when increafed in the proportion of the fine to radius*; 
and 2dly, that the difference between the logarithms of two fines, is 
lefs than the difference of the fines increafed in the proportion of thp 
lefs fine to radius, but greater than the faid difference of the fines in* 
creaf(^ in th<; proportion of the greater fine to radius f. 

* By this firft theorem, r being radius, the' logarithm' of the fine /, is between r f 

and — r; and therefore, when / differs hut little from r, the logarithm of t will be 



nearly equal to 
r — t 



r^tXr-^t 



; the arithmetical mean between the limits r— # and 



2/ 



r ; but ftill nearer to r — s \/ "^ 



the faid limits. 



r— t 
Or —^i/rsf the geometrical mean bctweoq 



t By this fecond theorem, the difference between the logarithms of the two fines S 

S J S — J ' 

and/, lying between the limits— ^r and -r— r, will, when thofe fmes differ hut 

little, be nearly equal to 



Sa — /2 s+/ X «— / ^u • -1 .-I 

r; — r or «i r, tljeir arithmetical mean ; 

e c 

or ^^*^^T ^ ';Tf/* the geometrical mean; or nearly =:g^ 2r, by fubftitutifjg, in 



th« laft 4cziomiiiator, { . Sf / for v* S/, to which it is nearly cqu^U 



Ifcnc?, 



L06ARI T H MS. 



45 



Hence, by the ift theorem, the logarithm of looooooo, the nulius 
or firft term in tlie firft table, being o, the logarithm of 9999099, tho 
2(f term, will be between i and i. 000000 1, and will therefore oe equal 
to I 'ooocooo j very nearly : and this will be alfe the common diiFerence 
of all the terms or proportionals in the firft table; and therefore by 
the continual addition of this logarithm, there will be obtained the 
logarithms of all thefe 100 proportionals : confequently 100 times 
the /aid firft logarithm, or the laft of the above fums, will give 
100.000005, for the logarithm of 9999900.000^950, the laft of the faid 
100 proportionals. 

Then, by the ad theorem, it eaiily appears that .0004950 is the dif- 
ference between the logarithms of 9999900.0004950 and 9990900, the 
iaft term of the firft table and the 2d term of the fecond table ; this 
then being added to the laft logarithm, gives 100.00050QO for the 
logarithm of the faid 2d term, as alfo the common difierence of the 
logarithms of all the proportionals in the ad table ; and therefore by 
continually adding it, there will be generated the logarithms of all the(e 
proportionals in the 2d table ; the laft of which is 5000.025, anfwer- 
mg to^999500i. 222027 the laft term of that table. 

Again, by the 2a theorem, th^ difference between. the logarithms 
of this laft proportional of the 2d table, and the 2d term in the firft 
column of the 3d table, is found to be 1.2235387 ; which being added 
to the laft lofi^arithm, gives 5001.2485387 for the logarithm of 
9995000, the faid 2d term of the 3d table, as alfo the common dif- 
ference of the logarithms of all the proportionals in the firft column of 
that table ; and this therefore, being continually added, gives all the 
logarithms of that firft column, the laft of which is 100024.97077, 
the logarithm of 9900473.5780, the laft' term of the faid column. 

Finally, by the 2d theorem again, the difference between the loga* 
rithms of tliis laft number and 9900000, the firft term in the 2d co- 
lumn, is 478.3502 ; which being added to the laft logarithm, gives 
100503.32 10 for the logarithm of the faid firft term in the 2d column, 
as well as the common difference of the logarithms of all the numbers 
on the fame line in every line of the table, namely, of all the i ft terms, 
of all the 2d, of all the 3d, of all the 4th, &c, terms in all the columns ; 
and which therefore, being continually added to the logarithms in the 
firft column, will give the correfponding logaritlims in all the other 
columns. 

And thus is compleated what the author calls the radical table, in 
which he retains only one decimal place in the logarithms (oxartifici^ 
ttls^ as he always calls them in his traft on the conftruAion), and four 
in the naturals. A fpecimen of the table is as here follows. 



Radical Table. 



Terms I 



ift Column.' ' 



z 
2 

3 

4 

5 

&c 

21 



Naturals. 
I cooco 30.0000 
9995000.0000 
9990002. 5C00 
99S5CC74987 
9980014.9950 

&c till 
0t50O47'^.<'»8o 



Artifici 

5001.X 

10002.5 

15003.7 

20005.0 

&c 



2d Column. 



N aturals. 
9900000.0000 
9895050.0000 
9890102.4750 

9885157.4*37 
9880214.S451 

9801468.842^ 



Artifici. 
100503.3 
105 504.6 
1 10505.8 
115507.1 
12050S.3 

&c 
200^28.2 



69th Column. 



Naturals. 
5048858.8900 
5046333.4605 
504381 1.2932 
5041289.3879 

5038768.7435 
&c 

4()c)86o9 40^4 



Artificials 
6834225.8 
6839227.1 
6844228r3 
68^9229.6 
6854230.8 

&c 
6Q342cr.8| 



Having 



4iSt CONSTRUCTION OF 

Having thus, in the moft eafy manner, compleatedthe radical table, 
by little more than mere addition and fubtraftion, both for the natu- 
ral numbers and logarithms ; the logarithmic fines were eafily deduced 
from it by means of the 2d theorem, namely, taking the fum and 
diflference of each tabular fine and the neareft number in* the radical 
table, annex 7 ciphers to the difference, divide the refult by the fum, 
and half the quotient will be the diiFerence between the logarithms of 
the faid numbers, liamely, between the tabular fine and radic^ nuinber ; 
confequently, adding or fubtrafting this difference, to Or from the 
given logarithm of tlie radical number, there will be obtained the loga- 
rithmic fine required. And thus tlie logarithms of all the fines from 
radius to the half of it, or from <p^ to 30', were perfeftedr 

Next, for determining' the fines of the remaining ?o degrees, he 
delivers two methods. In the firft of tbefe he proceeds in &is man^ 
ner : Obfervine that the logarithm of the ratio of 2 to i, or of half 
the radius, is 6931469.22, of 4 to one is the double of tliis, of 8 to % 
is triple of it, &q ; that of 10 to i is 23025842.34, of 20 to i is the 
fum of the logarithms of 2 and 10, and fo on by compofition for the 
logarithms of the ratios between i and 40, 80^ icx), 200, &c, to 
loooocxx) ; he multiplies ^ny given fine, for an arc Icfs than 30 de- 
grees, by fome of the& numbers, till he finds the product nearly 
equal to one of the tabular nun^bers ; then by means of this and the 
fecond theorem, the logarithms of this produft is found ; to which 
adding the logarithm that anfwers to die multiple abovementiohed, 
the fum is the logarithm fought. But the other method is ilill much 
eafier, and is derived from this property, which he demonftrates, 
namely, as half radius is to the fine of half an arc, fo is the cofine of 
the faid half arc to the fine of the whole arc ; or as f radius : fine of 
an arc : : qofine of the arc : fine of double the arc ; hence the loga- 
rithmic fine of an arc is found, by adding together the logarithms of 
half radius and of the fine of the double arc, and then fubtr^fting the 
logarithmic cofine from the fum. 

And thus the remainder of the fines, from 30° down to o, are eafily 
obtained. But in this latter way, the logarithmic fines for full one 
half .of the quadrant, or from o to 45 degrees, he obfcrves, may be 
derived ; the other half having already been mad? by the general 
method of the radical table, by one eafy divifion and addition or fub- 
tra£tion for each. 

I have dwelt the longer on this work of the inventor of logarithms, 
becaufe I have not feen in any author an account of his method of 
conftruding his table, although it is perfectly different from any other 
method uf^ by the later computers, and indeed almoft peculiar tp 
his fpeciesof logarithms. The whole of this work manifefts great 
ingenuity in the defigner, as well as much accuracy. But notwith- 
ilanding the caution he took to obtain his logarithms true to the 
neareft unit in the laft figure fct down in the tables, by extending the 
numbers in the computations to feveral decimals, and other means ; 
he had been difappointed of that end, either by the inaccuracy of his 
afiiiVant computers or tranfcribers, or through fome other caufe ; as 
the logarithms in the. table are commonly very inaccurate. It is re- 
markable 



LOGARITHMS. 47 

markable too that in thi$ traft on the conftruftion of the logarithms^ 
Lord Napier never calls them logarithms, but every where artifictMls^ 
as oppofed in idea to the natural numbers : and this notion of natural 
and artificial numbers I take to have been his firft idea of this matter^ 
and that he altered the word artificials to logarithms in his iirft book^ 
on the defcription of them, when he printed it, in the year 16 14, and 
that he would alfo have altered the word every where in this pofthu* 
mous work if he had lived to print it : for in the two or three pagea 
of appendix, annexed to the work by his fon from Napier's papers^ 
he again always calls them logarithms. This appendix relates to the 
change of the logarithms to that fcale in which i is the logarithm of 
the ratio of 10 to i, the logarithm of i, with or without ciphers, being 
c ; and it appears to have been written after Briggs communicated to 
him his idea of that change. 

Napier here in this appendix alfo briefly defcribes fome methods 
by which this new fpecies of logarithms may be conftru£ted. Having 
fiippofed o to be the logarithm of i , and I with any number of ciphers, 
as iopooocxx>oo, the logarithm of lO; he directs to divide this logarithm 
of 10, and the fucceflive quotients, ten times by 5, by which divifion$ 
there will be obtained thele other ten logarithms, namely aoooooocxx)^ 
4oooocxxx>, 80000000, 16000000, 3200000, 640600, laSooOy 25600, 
5120, 1024 : then this laft logarithm, and its quotients, beihg^divided 
ten times by 2, will give thefe other ten logarithms 512, 256, 128, 64, 
32, 16, 8, 4, 2, I. And the numbers anfwering to theie twenty loga- 
rithms, we are direfted to find in this manner ; namely, extcad the 
5th root of 10 (with ciphers), then the 5th root of that root; and fo 
on for ten continual extraftions of the ftb root ; fo ihall thefe ten roots 
be the natural numbers belonging to the firft ten logarithms, above 
foun<{ in continually dividing by 5 : Next, out of the lalt 5th root we are 
to extrad the fquare root, then the fquare root of tliis \m root, and fo 
on for ten fuccefiive extraAions of tiie fquare root ; fo fiiall thefe laft 
ten roots be the natural numbers correfiponding to the logarithms or 

; quotients arifing from the laft ten divinons by the number z. And 
rom thefe twenty logaridims, i, 2, 4, 8, 16, &c, and their natural 
numbers, the author obferves that other logarithms and their numbers 
may be -formed, namely by adding the logarithms and multiplying 
their correfpondent numbers. It is evident that this procefs would 
generate rather an antilogarithmic canon, fuch as Dod(on*s, than the 
table of Briggs ; and that the method would alio be very laborious, 
fince, befides the very troublcfome original extractions of the 5th roots, 
all tlie numbers would be very large, by the multiplication of which 
tlie fucceflive fecondary natural numbers are to be found. 

Our author next mentions another method of deriving a few of the 
primitive numbers and their logarithms, namely, by taking continually 
geometrical means, firft between 10 and i, then between 10 and this 
mean, and again between 10 and the lafft mean, and fo on ; and taking 
the arithmetical means between their correfponding logarithms. He 
then lays down various relations between numbers anud their logarithms, 
fucK a5 that the produds and quotients of numbers, anfwer to the fums 
and differences of their logarithms j and that the powers and roots of 

numbers, 



CdNSTRUCTIOM OF 

numbers, infwcr to tlic produfts and quotients of the logarithms by the 
index of the power or root, &c ; as alfo that, of any two numoers, 
whofe logarithms are given, if each number be raifed to the power 
denoted by the logarithm of the other, the two refults will be equal. 
He then delivers another method of making the logarirtims to a 
few of tlie prime integer numbers, which is well adapted for con- 
flrufting the common table of logarithms. This method eafily follows 
from what has been laid above, and .it depends on this property, that 
the logarithm of any number in tliis fcale, is i lefs than the number 
of places or figures contained in that power ofthe given number whofe 
exponent is iooooocx)000 or the logarithm of lo, at leaft as to integer 
■lumbers, for they really differ by a fraftion, as is fhewn by Mr. Briggs 
in his illuftiations of thefe properties, printed at the end of this 
appendix to the conftru£lion of logarithms. I Ihall here fet down one 
more of thefe relations, as the manner in which it is expreflcd is ex-* 
aftly iimilar.to that of fluxions and fluents, and it is this : Of any 
two numbers, as the greater is to the lefs, fo is the velocity of the 
increment or decrement of the logaritlims at the lefs, to the velocity 
of the increment or decrement of the logarithms at the greater: that 

mm • • • • 

IS, m our modern notation, as JT : I" : : jf to ;r, where x and^ are the 
fluxions of the logaHthms of X and T. 

Kepler's Con/Iru£iion of Logarithms. 

The logarithms of Briggs and Kepler were both printed the fame 
year, 1624; ^^^ ^ ^^ latter are of the fame fort as Napier's, I (hall 
here give the author's conftruftion of tliem, before we proceed to that 
of Briggs's. 

We have already (pa. 31 i^feq,) dcfcribed the nature and form of 
Kepler's logarithms, mewing that they are of the fame kind as Napier's, 
but only a little varied in the form of the table. It may alfo be added 
that, in general, the ideas which thefe two mailers had on this fubjeft, 
were of the fame nature, only it was more fiilly and methodically laid 
down by Kepler, who expanded, and delivered in a regular fcience, 
the hints that were given by the illuftrious inventor. The foundation 
and nature of their methods of conftrudion, are alfo the fame, but 
only a Httle varied in their modes of applying them. Kepler here firil 
of any treats of logarithms in the true and genuine way of the mea- 
lures of ratios, or proportions *, as he calls them, and that in a very 
full and fcientific manner : and diis method of his was afterwards fol- 
lowed and abridged by Mercator, Halley, Cotes, and others, as we 
ihall fee in the proper places. Kepler firfi ere£ls a regular and purely 
mathematical fyftem 0/ proportions, and tlie meafures of proportions, 



* Kcp]«r almoft always ufet XhAttrm fttpwiim infttad of fytfio, which I alfo ihall <!• 
in my account of his work, as weU as conform in expreiilioRS and notations to his other 

Gicnliaritits. It may alio b« here remarked, that I obfenre the fame pra6Hce in defcri- 
ng the works of other aochort, the better to convey the idea of their feveral methods 
and file. And this may fiervt to aiyotuiC for iome icemins iaequaUtics in the language 
of tbis hiftory. 

. treated 



LOGARITHMS. 49 

treated at coniiderable length in a number of propofitions, which are 
fully and chafteiy demonftratcd by genuine mathematical reafoning, 
and illuflrated by numerical examples. This part contains and de-* 
monftrates both the nature and tlic principles of the ftrufture of loga- 
ritbms, And in the fecond part he applies thofe principles in the 
a£tual conftruftion of his table, which contains only icx50 numbers 
and their Jogarithmsy in the form as we before defcribed ; and in this 
part he indicates the various contrivances made ufe of in deducing the 
logarithms of proportions one from another, after a few of the lead- 
ing ones had been firft formed by the general and more remote prin- 
ciples. He ufes the name logarithms^ given them by the inveotor, 
being the moft proper, as exprelTing the very nature and eflcnce of thofe 
artificial numbers, and containing as it were a definition in the very 
name of them i but without taking any notice of the inventor, or of 
the origin of thofe ufeful numbers. <^ 

i\s this tizdi is very curious and important in itfelF, and is befides 
very rare and little known, inftead of a particular defcription only, f 
fliall here give a brief tranflation of both tlie parts, omit;ting only the 
demonftrations of the propofitions, and fome rather long illuftrations 
of them. 

The book is dedicated to Philip, landgrave of Hefle, but is without 
either preface or introduction, and commences immediately with the 
fubjeft of the firft part, which is intituled I'he Demonjiratlon of the 
Struflure of Logarithms \ and the contents of it are as follows. 

Pojiulate I. That all proportions equal among themfelves, by what- 
ever variety of couplets of terms tliey may be denoted, are meafured 
or exprefled by the fame quantity. 

jfxiom I. It there be any number of quantities of the fame kind, the 
proportion of the extremes is undetftood to be compofed of all the 
proportions of every adjacent couplet of terms, from tlie firft to tlie 
laft. 

I Propq/stion- The mean proportional between two terms divides 
the proportion of thofe terms into two equ^l proportions. 

Axiom 2. Of any number of quantities, regularly increafing the 
liieans divide the proportion of the extremes into one proportion 
more than the number of the means. 

Poftulate 2. That the proportion between any two terms, is divifi- 
ble into any number of parts, until thofe parts become lefs than any^ 
propofed quantity. 

An example of this feAion is then inferted in a fmall table, in dividing the proportioa 
which is between lo and 7 into 1073741824 equal parts, by as many mean proportionals 
wanting one, namely, by taking the mean prop9rcion9l between 10 and 7, then the meaa 
between iq and this mean, and the mean between to and the laft, and fo on for 30 means, or 
3oextraAions of the fquare root, the laft or ^oth of which roots is 99999999966782056900 
and the 3cth power of 2, which is 1073741 824, (hews into how many parts the proportion' 
between 10 and 7, or between 1000 &c and 700 &c, is divided by 1073741824 means 
each of which parts is equal to the proportion between 1000 &c, and the 30th mean 009 
Ice. that is the proportion between looo &c and 999 &c, is the 1073741824th pait of tne 
proportion between 10 and 7. Then by afTuming the fmall difference 00000.00003.32179, 
43100, for the meafure of the very fmall element of the proportion of 10 to 7, or (or the 
itieafore of the proportion of 1000 &c to 999 &c, or for the logarithm of this laft term, 
and mnliiplying it by 10737418x4, the number of parts, the produft will give 35667.49481 
37 »22. 14400 for the logahchin «f tbt left term 7 or 700 lc«« 

H Poftulati 



50 . CONSTIiUCTIO^ Of 

Pojlulati 5. That the extremely fmall quantity or clement of a pt(f* 
portion, may be meafured or denoted by any quantity whatever 5 ad 
for inftance, by the difference of the terms of tliat element. 

2 Propofition, Of three continued proportionals, the difference of 
the two firft has to the difference of the two latter, the fame propor- 
tion which the firft term has to the 2d, or the 2d to the 3d. 
. 3 Prop. Of any continued proportionals, the greateft terms have the 
greateft difference, and the Icaft terms the leaft. 

4 Prop. In any continued proportionals, ?f the difference of the great- 
eft terms be made the meafure of the proportion between ti^rriy the dif^ 
fercnce of any other couplet will be lefs than tlae true meafure of their 
proportion. 

SProp. In continued proportionals, if the difference of the greateft 
terms be made the meafure of their proportion, then the proportion of 
tlie greateft to any other term will be greater than their difference. 

6 Prop. Jn continued pioportionals, if the difference of the greateft 
term and any one of the lefs, taken not immediately next to it, be made 
the meafure of their proportion ; then the proportion which is between 
the greateft and any other term greater than the one before taken, will 
be lefs than the difference of thofe terms ; but the proportion which is 
between the greateft term and any one lefs than that firft taken, wiU 
be greater than their difference. 

7 Prop. Of any quantities placed according to the order of their 
magnitudes, if any two fuccefhve proportions be equal, the three fuc- 
ceffive terms which^ conftitute them, will be continued proportionals^ 

8 Prop. Of any quantities placed in the order of their magnitudes,, 
if the intermediates lying between any two terms, be not among the 
mean proportionals which can "be interpofed between the fakl two 
terms, then fuch intermediates do not divide the proportion of thofir 
two terms into commenfurable proportions. 

Beiides the demonftrattODS| as ufual, feveral definitions are bere given ; as of commenfa* 
nble proportions, &c. 

gProp. When two cxpreiBble lengths are not to one another 29 
two figurate numbers of the fame fpecies,.fuch as two fquares, or two 
xubes ; there cannot fall between them other expreffiblc lengths, 
which fhall be mean proportionals, and as many in number as that 
fpecies requires, namely, one in the fquares^ two in the cubes» three 
in the biquadrats, &c. 

loProp. Of any exprcflible quantities, following in the order oi; 
their magnitudes, if the two extremes be not in the proportion of two 
fquare numbers, or two cubes, or two other powers of the fame kind ; 
none of the intermediates divide the proportion intocommenfurables. 

1 1 Prop. All the proportions, taken in order, which are between 
expreflible terms that are in aritlimetical proportion, are inconuoeii* 
furable to one another. As between 8, 13, iS. 

12 Prop. Of any quantities placed in the order of magnitude, if thei 
difference of the greateft terms he made the meafure of their propor- 
tion, then the difference betwecp any two others will be lefs than tho. 
meafure of their proportion ; and if the difference of the two leafk 
terms be made the meafure of their pjoportiouj theu (he differences of 



LOGARTTIiMS. ' s\ 

tlie reft will- be greater than the meafure of the proportion between 
their terms. 

Carol. If the nieafure of the proportion between the greatcft, ex- 
ceed their difference, then the proportion of this meafure to the 
faid difference, w^l be lefs than that of a following meafure to the 
difference of its terms. Becaufe proportionals have the fame ratio. - 

13 Prop. If three quantities follow one another in the order of mag- 
nitude ; the proportion of tlie two laft will be contained in the propor- 
tion of the extremes, a lefs number of times than the difference of 
the two leafi is contained in the difference of the extremes : And on 
the contrary, the proportion of the two greateft will be contained in 
the proportion of the extremes, oftener than the difference of the 
former is contained in that of th^ latter. 

CoroL Hence if the difference of the two greater be equal to the. 
difference of the two lefs tfcrms, the proportion between the two 
greater will be lefs than the proportion between the two lefs. 

14 Prop. Of three equidlfferfent quantities taken in order, the pro- 
portion between the extremes is more than double the proportioa 
between the two greater term^. 

CoroL Hence it follows, that half the proportion of the extremes, 
is greater than the proportion of the two greateft terms, but lefs than 
the proportion of the two leaft. 

15 Prop. If two quantities conftitute a proportion, and each quan- 
tity be leffened by half the greater j the remainders will conftitute a 
proportion greater than double the former. 

16 Prop. The aliquot parts of incommenfurable proportions, are 
jncommenfurable to each other. 

17 Prep. If one thoufand numbers follow one another in the natu- 
ral order, beginning at 1000, and differing all by unity, viz. looo, 
999, 998, 997, &c ; and the proportion between the two 'greateft 1000, 
999, by continual bifedlion, be cut into parts that are fmaller than 
the excefsof the proportion between the rtext two 999, 998, over the 
faid proportion between the two greateft 1000, 999; and then fqr 
the meafure of that fmall element of the proportion between loco, 
999, thexebc taken the difference of icooand that mean proportional 
which is the other term of the element. Again, if the proportion be- . 
tween 1000, 998 be likewife cut into double the niimb^ of parts 
which the former proportion between icod, 999 was cut into; and 
then for the meafure of the fmall element in this divifion, be taken the 
difference of its terms, of which the greater is 1000. And, in the 
iame manner, if the proportion of 1000 to the following num- 
bers, as 997, &c, by continual bifeflion, be cut into particles of fuch 
magnitude, as may be between | and^ of the element ariling from 
the fedlion of the firft proportion between 1000 and 999 ; the meafure 
of each dement will be given from the difference of its terms. Then, 
this being done, the meafure of any one of the 1000 proportions, 
will be compofed of as many meafures of its element, as there are of 
thofe elements in the faid divided proportion. And all thefe mea- 
fures, for all the propoitioASj will be fttfficiently ^%s^£\ for the niceft 
calculations, ^ 

Hi All ' 



CONSTRUCTION OF 

All thefe fe^ions and me'afures of proportionc: are performed in the manner of th^^ 
described at po^uLite z, and the opcrationvis ibutidaatly explained by numerical calc^- 
tations. 

liProp. The proportion of any number to the firft term looo being 
known ; there will alfo be known the proportion of the reft of the 
numbers in the fame continued proportion, to the faid firft term. 
jSo from' the known proportion between lOOO and goo, 

there is alfo Renown the proportion of lOOo to 8iO| and to 729 ; 
And from 1000 to 8ooy Mo tooo to 640, and to jX2 ; 
And from 1000 to 700, alfo 1000 to 490, and to 343 ; 
And from 1000 to 60O9 alfo 1000 to 360, and to 216 ; 
And from 1000 to 500,, alfo 1000 to 250, and to 125* 

' CoroL Hence arifes the precept for fquarings cubing, &c ; as alfo 
for extradling the fquare root, cube root, &c, out oJF the firft figures 
of numbers.. For it will be, as the greateft number of the chiliad as 
a denominator, is to the number propofed as a numerator, fo is this 
10 the fquare of the fradlion, and fo is this to the cube. 

igProp. The proportion of a number to the firft, or lOOO, being 
known ; if there be two other numbers in the fame proportion to each 
other, then the proportion of one of thefe to tpoo being known, there 
will alfo be known the proportion of the other to the lame lOoo* 

CcfroL z. Hence from the 15 proportions mentioned in prop. 18, 
will be known 12Q others below 1000, to the fame iQop. 

For fo many are the proportions, equal to fome one or o(her of the faid 15, that are 
among the other i?^teger numbers which are lefs than looo. 

CoroL 2* Hence arifes the method of treating the RuIe<*of- Three, 
when icoo is one of the given terms. 

For this is effe^ed by adding to, or fubtradting from« obch other, the mea(\ires of th« 
two proportions of icoo to each of the other two given numbers^ according as 1000 is, or 
is noCy the firft term in the Rule>of-three. 

20 Prop. When four numbers are proportional, the firft to the fe- 
cond as the third to the fourth, and the proportions of lOOO to each 
of the three former are known, there will alfo be known the propor<? 
tion of locxi to the fourth number. 

CoroL I. By this means other chiliads are added to the former. 

CorcL 2. Hence arifes the method of performing the Rule-of- three, 
when 1000 is not one of the terms. Namely, from the fum of the 
meafures of the proportions of icoo to the fecond and third, take that 
of idco to the firft, and the remainder is the meafure of the proportioA 
of 1000 to the fourth term. 

Definition. The n^eafure of the proportion between lOOO and any 
lefs nqmber, as before defcribed, and expreiTed by a number, is fet 
oppofite to that lefs iiumber in the chiliad, and is called its loga<p- 
RITHM, that is, the number (cepi0/ud<) indicating the proportion 
(xoyof) whi<^h 1090 bears to that number, to which the logarithm 19 
annexed.. 

izi Prop, If the firft or greateft number be made the radius of a 
'circle, or Jinus totus \ every lefs number, cpnfidered as the cofine of 
fome arc, has a logarithm greater than the verfed fine of, that arc, 
but lefs than the difference between the radius and fecant of the arc» 
cxcej)t only in the term next after the radius, or greateft term^ t^e 
logarithm of wbicl) by the hyiigthi^s is made e<}ual to the verled fine. 



tO&ARITHMS. 







That 1$, if C D be mado the loj^arithm of A Cy or the iqeafure of 
the proportion of AC to A D ; then the meafure of the proportion 
qf A B to A D« that is the logarithm of A B, will be greater than 
B D, but lefs than £ P. And this is the fame as Napier's firft rule 
in page 44, 

A 

22. Prop* The fame things being fuppofed ; the fum of the verfcd 
fine and exccfsof the fecant over the radius, is greater than double 
the logarithm of the cofine of an arc, 

CoroL The log. cofine Is lefs than the arithmetic mean between 
the verfed fine and the excefs of the fecant. 

Precept I. Any fine being found in the canon of fines, and Its de- 
fed below radius to the excefs of the fecant above radius; then (hall 
the logarithm of the fine be lefs than half that fum, but greater than 
the faid dtkSt or coverfed fine. 

Let there be the fine ^ 99970.1490 of an arc: > ' 

Its defedl below radius is 19.85 10 the covers, and lefs than logarithm fine ; 
Add the e3(cefs of the fecant 29.8599 

Suin 59.7109 
its half or a9>8555 greater than the logarithm* 

Therefore the logarithm is between { j^;3^^°/"'* 

Precept 2. The logarithm of the fine being found, you will al(b 
find nearly the logarittim of the round or integer number which is next 
lefs than your fine with a fraftion, by adding that fradlional excefa to 
the logarithm of the faid fine. 

Thus the log^rilhip of the fine 99970.149 is found to be about 19.854; if now the loga* 
rithm of the round number 99970.C00 be required, add 149 the fractional part of the fine 
lo its logarithmi obferving the point, thus 29.8 54 T. ^ , . 

i^Q J IS the logarithm of the round number 

the fum 5^^9997oaoooneariy. 

23. Prop: Of three equidifferent quantities, the meafure of the 
proportion between the two greater terms, with the meafure of the 
jproportion between the two lefs terms, will conftituie a proportion, 
which will be greater than the- proportion of the two grater tenns, 
bi|t lefs than the proportion of the two leaft. 

Thus if AB, AC, AD be three quantities having the equal - ' ■ ' it A K 

differences BC, CD ; and i£ the meafure of the proportion of "^ B C i> 

AD, AC be cd, and that of AC, AS be be j then the propor- J ' ' 

tion of cd to cb will be greater than the proportion of AC to oca 
^D, but leis than the proportion of A B to AC. 

24 frop. The faid proportion between the two meafures, is ]e(8 
^an half the proportion between the extreme terms. That is, the 
proportion between be, cd, is lefs than half the proportion between 

/IB, AD. 

Corol. Since therefore the arithmetical mean divides the proportion 
Jnto unequal parts, of which the one is greater, and the other lefs, 
than half the whole; if it be enquired what proportion is between 
thefe proportions^ the anfy^er is^ that i( is a little lefs than the faid 

w^ ^ . 



%^ CONSTRUCTION OF 

jIh Example tf finding marly the limits^ greater and lefs^ to the meafurc 

of any propofed prgporiion. 

It being known that the meafure of the propoition between xooo and 900 is 1053^.05^ 
required the meafure of the proportion 900 to 800, where the terms 1000, 900, 800, have 
equal differences. Therefore as 9 to i o fo 105 36.05 to 1 1 706.72, which is lefs than i z 7^8.30 
the meafure of tiie proportion 9 to 8. Again, as the mean proportional between S and 10 
(which is 8.9442719) is to 10 fo 10536.05 to 11779.66, which is greater than the meafure 
of Che proportion between 9 and 8. 

Axiom, Every number denotes an expreffible quantity. 

25 Prop, If the 1000 numbers, differing by i, follow one another 
in the natural order; and there be taken any two adjacent numbers, 
as the terms of foine proportion ; the meafure of this proportion will 
be to the meafure of the proportion between the two greateft terms of 
the chiliad, in a proportion greater than that which the greateft term 
1000 bears to the greater of the two terms firft taken, but lefs than 
the proportion of 1000 to the lefs of the faid two fele<Sted terms* 

So of the TOGO numbers taking any two fuccellive terms, as 501 and 500, the logarithm 
(Df the former being 691 14.92, and of the latter 69314.72, the diffierence of which is 299.80, 
Wherefore by the definition, the meafure of the proportion between 501 and 500 isi99.8o« 
In like manner, becaufe the logarithm of the greatefl term 1000 is a, and of the next 
999 is 100.05, the difi'erence of thefe logarithms, and the meafure of the proportion be^* 
tween 1000 and 999, is 100.05. Couple now the greateft term xooo with each of tho 
feleAed terms 501 and 500 ; couple alfo the meafure 199.80 with the meafura 1 00,05; fo 
Ihall the proportion between i99«3o and i»c.o5 be greater than the proportioo betweea 
|oooand 50I9 but leis than the proportion between loooand 500. 

CoroL I. Any number below the firfl 1000 being propofed, as alfo 
its logarithm ; the differences of any logarithms antecedent to that 
propofed, towards the beginning of the chiliad, are to the firft loga- 
rithm (viz. that which is ailigned to 999)9 in a greater proportion than 
1000 to the number propofed ; but of thoie which follow towards thq 
)aft logarithm, they are to the fame in a lefs proportion. 

CoroL 2» By this means the places of the chiliad may eafily be filled 
up, which have not yet had logarithms adapted to them by the former 
propofitions. 

26 Prop* The difFerence of two logarithms, adapted to two adjacent 
numbers, is to the difference of thefe numbers, in a proportion greater 
than 1000 bears to the greater of thofe numbers, but lefs than that o( 
I PCX) to the lefs of the two numbers. 

This 26 prop, is the (ame as Kapler's fecond rule at page 44. 

7.y Prop, Having given two adjacent numbers of the xooo natural 
numbers, with their logarithmic indices, or the meafures of the pro* 
portions which thofe abfolute or round numbers conftitute with loob 
the greateft ; the increments or differences of thefe logarithms will be 
to the logarithm of the fmallelement of the proportions, as the fecants 
of the arcs whofe cofines are the two abfolute numbers, is to thegreat- 
eft number, or the radius of the circle : fo that, however, of the faid two 
fecants, the lefs will have to the radius a lefs proportion, than the pro- 
pofed difFerence has to the firft of all, but. the greater will have a 
greater proportion, and fo alfo will the mean proportional between th^ 
faid fecants have a greater proportion. 



X.06AIt.ITHMS. 



is 




^us if B Cy C D be equal, alfo ^<f the logarithm of A B, and cd 
the logarithm of A C ; thea the proportion of ^ c to cd will be 
greater than the proportion of AG to A D, but lefs than that of 
A F to A D> and alfo lefs then that of the mean proportional be- 
tween A F and A G to A D. IL 

CoroL I. The fame obtains alfo when the two terms differ, not 
only by the unit of the fmall element, but by another unit which may 
be ten fold, a hundred fold, or a thoufand fold of that. 

CoroL 2.. Hence the differences will be obtained fufficiently exa^^^ 
efpecially when the abfolute numbers are pretty large, by taking the- 
arithmetical mean between two fmall fecants, or (if you will be at 
the labour) by taking the geometrical mean between two larger fecants* 
and then by continually adding the differences, the logarithms will 
be produced. 

CoroL 3. Precept. Divide the radius by each term of the affigneJ 
proportion, and the arithmetical mean (orflill nearer the geometrical 
mean) between the quotients will be the required increment, which 
being added to the logarithm of the greater term^ will give the loga- 
rithm of the lefs term. 

Example* 

Let there be given the logarithm of 700, viz. 35667.4946, to find the lofantbm to 6^^ 

Here radius divided by 700 gives 1428571 &c. 
and divided by 699 gives 1430672 &c. 
•Che arithmetic mean is 142.96a 
which added to 35667494S 

gives the logarithm to 699 35810.456S 

CoroL 4. Precept for the logarithms of fines. 

The increment between the logarithms of two fines, is thus found : 
fitid the geometrical mean between the cofecants, and divide it by the 
difference of the fines, the quotient will be the difference ot th^ 
logarithms. 

ExMTiple. 



c^ i' fine 2909 cofec. 343774682 

fine 5818 cofec. 171887319 

dif. 2909 geom. mean 2428 nearly* 



o a 



The quotient 80000 exceeds the requkw 
ed increment of the logarithms, becaufe thf 
fecants are here fo large. 

j/ppendix. Nearly in the fame manner it may be (hewn, that the 
fecond differences are in the duplicate proportion of the firft, and the 
third in the duplicate of the fecond. Thus for inftance, in the be* 
ginning of the logarithms the firfl difference is loo.coooo, viz. equal 
to the difference of the numbers looooo.ooooo and 99^00.00000 ; the 
fecond, or difference of the differences, lOQoo; the third 20. Again 
after arriving at the number 50000.00COO, the logarithms have for a 
difference 200.00000, which is to the firft difi^(?rence, as the number 
iocxxx>.ooooo to 50000.00COO9 but the fecond difference 1840000^ in 
which loooo is contained four times; and the third 328, in which 20 
is contained fixteen times. But fince in treating ot new matters we 
labour under the want of proper words, wherefpre leaft we ihould be-* 

4Mnq tgp obfcure^ the demofiilratipa i$ omitted untried. 

28 Pr$f. 



^ CQWSTRlJcfldl^OF 

28 Prop* No number exprefles exadUy the meafure of the propdr- 
tioh, betwcea two of the IQOO numbers, conllituted by the foregoing 
method. 

29 Prop. If the meafures of all proportions be expreffed by numbers 
Or logarithms; all proportions will not have afligned to them their due 
portion of meafure, to the utmoft accuracy. 

-30 Prop. If to the number' icxxD, the grcateft of the chiliad, be 
fe^rred others that are greater than it, and the logarithm of jooo be 
made o, the logarithms belonging to thofe greater numbers will be 
negative. 

This concludes the firft or fcientific part of the work, the princi- 
ples of which Kepler applies, in the fecond part, to the adual conftruc- 
tion of the firft lOoo logarithms, which is pretty minutely defcribed. 
This part is entituled Avery compendious method of conJlru£iing the Chi* 
Had of Logarithms i and it is not improperly To called, the method 
being very concife and eafy* The fundamental principles are briefly 
thefe : That at the beginning of the logarithms, their increments or 
difFerences are equal to thofe of the natural numbers: that the natural 
numbers may be confidered as the decreaiing cofines of increadng 
arcs : and that the fecants of thofe arcs at the beginning have the fame 
differences as the cofines, and therefore the fame differences as the 

' logarithms. Then, fince the fecants are the reciprocals of the cofines, 
by thefe principles and the third corol. to the twenty- feventh pro- 
|x>fition, he eftablifhes the following method of conftituting the 100 
£rft or fmalleft logarithms to the 100 largeft numbers, 1000, 999, 
998, 997, &c to 900. viz. Divide the radius 1000, increafed with 

• feven ciphers, by each of thefe numbers feparateiy, difpofing the quo- 
tients in a table, and they will be the fecants of thofe arcs which 
bave the divifors for their cofines; continuing the divifion to the 
Sth figure, as it is in that place only that the arithmetical and geo 
metrical means differ. Then by adding fuccefSvely the arithmetical 
means between every two fucceffive fecants, the fums will be the feries 
of logarithms* Or by adding continually every two fecants^ the 
fucceflive fums will be the feries of the double logarithms. 

Befides thefe 100 logarithms, thus conftruded, he conflitutes two 
ethers by continual bi(e£lion, or extractions of the fquare root, after 
the manner defcribed in the fecond poflulate. And nrfl he finds the 
logarithm which meafures the proportion between 100000.00 and 
97656.25> which latter term is the third proportional to 1024 and 
1000, each with two ciphers ; and this is efFe<Sed by means of twenty- 
four continual extradions of the fquare root, determining the greatefl 
term of each of twenty-four clafTes of mean proportion's ; then the 
difference between the greateft of thefe means and the firfl or whole 
number. 1000, with ciphers, being as often doubled, there arifes 
z2Ji*6$26 for the logarithm fought, which made negative is the lo- 
garithm of 1024. Secondly, the like procefs is repeated for the pro^ 
portion between the numbers 1000 and 500, from which arifes 
^9314*7193 fo^ ^hc logarithm of 500 ; yrhich he alfo calls the logar- 
ithm of duplication, being the meafure of the proportion of 2 to i. 

Then 



LOGARITHMS. 



57 



Then from the foregoing he derives all the other logarithms in the 
chiliad, beginning with thofe of the prime numbers i, 2, 3, 5, 7, &€« 
in the firft 100. And firft, fmce 1024, 512, 256, 128, 64, 32, 16, 8, 
4, 2, I, are all in the contin.ued proportion of lodo to 500, therefore 
the proportion of 1024 to i is decuple of the proportion of IOCX3 to 
500, and cOnfequently the logarithm of i would be decuple of the 
logarithm of 500, if Q were taken as the logarithm of 1024 ; but fince 
the logarithm of 1024 is applied negatively, the logarithm of i muil: 
be diminifhed by as much : dimifii(hing therefore 10 times the logar^ 
ithm of 5CX5, which is 693147. 1928, by 2371.6526, the remainder 
690775.5422 is the logarithm of i, or or iqo.oq what is fet down ia 



Nos, 



100 

10 

I 

.1 

.01 

.001 

0001 



the table. 

And becaufe i, 10, lOO, looo, are continued pro- 
portionals, therefore the proportion of 1000 'to i is 
triple of the proportion of lOQO to lOO, and confe- 
quently j of the logarithm of i is to be put for the 
logarithm of XOO, viz. 230258.5141, and this is 
aifo the logarithm of decuplication, or of the pro- r 
portion of iQ to i. And hence multiplying this logarithm of 100 
fucceffively by 2, 3, 4, 5, 6, and 7, there arife the logarithms to the 
/lumbers in the decuple proportion, as in the margin. 
• Alfo if the logarithm ot duplication, or of the jLog.of i 
proportion of 2 to i, be taken from the logarithm of 2 to 



Logarithms. 

230258.5141 

460517.0282 

690775.5422 

921034.0563 

1151292.5703 

1 38 1 55 1.0844 

161 1809.5985 



log. of 2 
log« of 10 

of 5 to I 
. !<«• of 5 



690775-54** 
69314.7193 



621460.8229 

4605 17.028 1 



160943.7948 



529331.7474 



of I, there will remain the logarithm of 2 ; and 
from the logarithm of 2 taking the logarithm of 
lo, there remains the logarithm of the proportion 
of 5 to I ; which taken from the logarithm of i, 
there remains the logarithm of 5. See the margin* 

For the logarithms of other prime numbers he has recourfe to thofe of 
fon^e of the firft or greateft century of numbers, before found, viz. of 999, 
998, 997, &c. And firft, taking 960, whofe logarithm is 4082.2001 ; 
then by adding to this logarithm the logarithm of duplication^ there 
will arife (he feveral logarithms of all thefe numbers, which are in 
duplicate proportion continued from 960, narneiy 480, 240, 120, 60, 
3D, 15. Hence the logarithm of 30 taken from the logarithm of 10, 
leaves the logarithm of the proportion of 3 to i ; which taken from 
the logarithm of i, leaves the logarithm of 3, viz. 580914.3106. 
And the double of this diminifhed by the logarithm of i, gives 
^71053.0790 for the logarithm of 9. 

Next, fronv the logarithm of 990, or 9 x 10 x 11, which is 
1^5.0331, he finds the logarithm of 11, namely, fubtra£l the fum 
of the logarithms of 9 & le from the fum of the logarithm of 990 and 
double the logarithm of i, there remains 450986.0106 the logarithn^ 
ofii. 

Again, from the logarithm of 980, or 2 x 10 x 7 x 7, which i^ 
^p20.27ii, he finds 496184.5228 for the logarithm of 7. 

And from 5 129.3303 the logarithm of 950 or 5 x lO X 19, he find)} 
296J31.6392 for the logarithm of 19, 



>;» CONSTRUCTIONQF 

In like manner the logarithm 

to 998 or 4 X 13 X 19, gives the logarithm of 13 j 
to 969 or 3 X 17 X 19, gives the logarithm of 17 ; 
to 986 or 2 X 17 X 29, gives the logarithm of 29 ; , 
to 966 or 6 X 7 X 23, gives the logarithm of 23; 
to 93© or 3 X 10 X 31, gives the logarithm of 31. 
And fo on for all the primes below ico, and .for many of t}^^ 
pripies in the other centuries up to 900, After which he direftq 
to find the logarithms of all numbers compofed of thefe, by the pro-^ 
per addition and fubtradion of their logarithms, namely, in finding 
the logarithm of the produ£l of two numbers, from the fum of the 
logarithms of the two fa<ftors take the logarithm of i, the remainder 
is the logarithm of the product. In this way he (hews that the logar- 
ithms of all numbers under 5Q0 may be derived, except thofe of the 
following 36 numbers, namely 127, 149, 167, 173, 179, 211, 223, 
asi. 257, 263, 269, 271, 277, 281, 283, 293, 337, 347, 349, 353, 

3S9» 367* 373> 379i 383v389> 397» 40i, 4P9» 4*9' 42i, 43i> 433» 
439, 443, 449. Alfo befides the compoiite numbers between 500 

and 90O9 made up of the products of fome numbers whofe logarithms 
have been before determined, ;there will be 59 primes not compofed 
of them; which with the 36 above mentioned, make 95 numbers in 
all not compofed of the produdls of any before them| and the logar- 
ithms of which he direds to be derived in this manner ; namely, by 
confidering the dilFerences of the logarithms of the numbers inter- 
fperfed among them ; then by that method by which were conftitutcd 
the diiFerences of the logarithms of the fmalleft 100 numbers in a con^ 
tinued feries, we are to proceed here in the difcontinued feries, that 
is, by prop. 279 coroK 3, and efpecially by the appendix to it, if it 
be rightly ufed, from whence thoie diiFerences will be very eaiUyr 
fupplied. 

This clofes the fecond part, or the actual coftrudion of the logar- 
ithms; after which follows the table itfelf, which has been before 
defcribed, pa. 31. Before I difmifs Kepler's work, however, it may 
not be improper in this place to take notice of an erroneous property 
laid down by him in the<appendix to the 27th prop, jqft now referred 
to; both becaufe it is an error in principal, tending to vitiate the 
practice, and becaufe it ferves to (hew that Kepler was unacquainted 
with the true nature of the prders of differences of the logarithms^ 
notwithftanding what he fays above with refpeA to the conftru<^ioa 
of them by means of their feveral orders of differences, and that con-r 
fequently he has no legal claim to any fliare in the difcovery of tho 
differential method, knpwn at that time to Briggs, and it would feem 
to him alone, it being publifhed in his logarithm$ in the fame year 
1624, as Kepler's book, together with the true nature ofthe logar- 
ithmic orders of differences, as we fhall prefently fee in the following 
account of his works. Now this error of Kepler's, here alluded to^ 
is in that cxpreilion where he fays the third differences are in the 
duplicate ratio of the fecond differences, like as the fecond differences 
are in the duplicate ratio of the firft ; or, in other words, that the 
third differences are as xht /quarts oi thp fecond cjifferences^ as well as 

■^ * t^9 



LOGARITHMS, 



59 



t}ii§ fecond difFerences as the fquares of the firft; or that the third 
t^ifFerences are as the fourth powers of the firft differences. Whereas 
ill truth the third differences are only as the cubes of the firft differen- 
ces. Kepler fcems to have been led into this errcfr by a miftake in his 
tiumber^, viz. when he fays in that appendix, that the third difference 
is 328, in which 20 is contained if) times; for when the numbers are 
accurately computed, the third difference comes out only 161, in 
"Which therefore 20 is contained only 8 times, which is the cube of 
2, the number of times the one firft difference cpntains the other. 
It would herice feem that Kepleir had haftily drawn the above errone- 
ous principle from this one numerical example, or little more, falfe as 
it is : for had he made the trial in many inftances, sllthough errone- 
t)ufly computed, they could not eafily hive been fo uniformly fo, as 
to afford the fame falfe conclufion. And therefore from hence, and 
ivhat he fays at the conclufion of that appendix, it may be infered 
that he either never attempted the demonftration of the property in' 
queftion, or elfe that he found himfelf embarrafed with it, an4 un- 
able to accompliih it^ and therefore difpatchcd it in the ambiguous 
hianner in which it appears. 

But it may eafily be fhewn, not only that the third differences of the 
logarithms at different places, are as the cubes of the firft differences ; ' 
but, in general^ that the numbers in any one and the fame order of 
^differences j at different places, are as that poVver of the numbers in 
the firft differences, whofe index is the fame as that of the order ; or 
that the fecond, third, fourth, .&c, differences, will be as the fe- 
cond, third, fourth, &Ci powers of the firft differences. For the 
feveral orders of differences, when the abfolute numbers differ by. 
indefinitely fmall parts, are as the feveral orders of fluxions of the 

Jogarithms ; but if x be any number, then ~ is the fluxion of the 

logarithm of Xj to the modulus m^ and the fecond fluxion, or the flux-. 

ion of this fluxion^ is ^, fince x is conftant ; and the third^ 

fourth, &c, fluxions, are ^^, — iJ^, &cj that is, the firft, ft- 
condj third, fourth, fifth, fixth, &c, orders of fluxions, are equal to 

X 1 X* 1.2 X^ 

the modulus m multiplied into each of thefe terms, — » — ^» j-» 

X XT X 

^ d—^ 1-3 — , ^-2 — > &c, where it is evident 

that the fluxion of any order, is as that power of the firft fluxion, Whofe 
index is the fame as the number of the order. And thefe quantities 
would adlually be the feveral terms of the differences themfelves, if 
the differences of the numbers were indefinitely fmall. But they vary 
the more from them, as rhe differences of the abfolute numbers differ 
from ', or as the faid conftant numerical difference I, approaches 
towards the value of *• the number itfelf. However, upon the whole, 
the feveral orders vary proportionably, lb as ftill fenfibly to prcferve 

la th? 



6* CONSTRUCtlOIt OP 

the fame analogy, namely that two nth difFercnces are ill pfoportJorf 
sis the ;ith powers of their refpe£live firft differences. 

Of Briggs^s Conjlru^iqtt dfhis Logarithms. 

Nearly accor3ing to the methods defcribed in page 47, Mr. Briggs 
tonftru(£ied the logarithms of the prime numbers^iis appears from hisr 
relation of this buftnefs in the Arithmetica Logarithmica^ printed in 
1624, where he details^ in an ample manner, the whole eonftrudion 
and ufe of his logarithms. The work is divided into thirty-two chap- 
ters oi'feftions. In. the firft of thefe^ logarithms ina general fenfeare 
defined, and fome properties of them illuftrated. In the fecond chap- 
ter he remarks, that it is moft convenient to make o the logarithm 0/ 
I ; and on that fiippofition he exemplifies thefe following properties, 
namely, that the logarithms of all numbers are either the indices of 
powers^ or proportional to them ; that the fum of the logarithms of 
two or lilore factors, is the logarithm of their produd y and that the 
difference of the logarithm- of two numbers, is the logarithm of their 
quotient. In the third feAion he ftates the other afTumption which is 
neceffary to limit his fyftem of logarithms, namely, making i the 
logarithms of 16, as that which produces the moft convenient form of 
logarithms ; He hence alfo takeft occaiion to fliew that the powers of 
10, namely icx),iooo, &c, are the only numbers which can have 
rational logarithms. The fourth fe6iion treats of the cKarafteriftic 5 
by which name he diftinguiflies the integral, or firft part, of a loga-*' 
rithraT towards the left-hand, which exprelTes onelefs than the num- 
ber of integer peaces or figures in the number belonging to that logar- 
ithm, or how far the firft figure of this number is removed from the 
j^lace of units ; nameiyi that o is the charaSeriftnc of the logarithms 
iof all numbers from i to to ; and i the charaSeriftic of all thofe from 
JO to 100 J and 2 that of thofe from 100 to 1000 ; and fo on. 

He begins the fifth chapter With rcmarlcing, that his logarhhmis 
may chieny be conftru£l:ed by the two-methods which were mentioned 
\>y Napiet, as above related, and for the fake of which he here pref- 
iniifes feveral Umniataj concerning the powers of numbers and their 
indices, and how many places or figures are in the ptodudts of lum- 
bers, obferving that the prodii6l of two numbers will confift of as 
many figures as there are in both fadlors, unlefs perhaps the produft 
of the firft figures in each fadlor be exprefled by one figure only, 
which often happens, and then commonly there will be one figure in 
the product lefs than in the two fa£lors ; as alfo that, of any two df 
the terms, in a ieries oCgeometricals, the refults will be equal by 
raifing each term to the power denoted by the index of the other ; or 
any number raifed to the power denoted by the logarithm of the 
other, will be equal to this latter number raifed to the power denoted 
by the logarithm of the former ; and confequently if the one number 
be 10, whofe logarithm is i with any number of ciphers, then any 
Xiumber raifed to the power whofe inoex is 1000 &c, or the logarithm 
of 10, yifill be equal 19 10 railed tq the pow^r whofe index is the lo- 

garithnpL 



LOOARl T HMg. 



«i 



^fitllm of that number; th^t is, the logarithm of any number m 
this (bale,., where i is the logarithm of lo, is the index of that 
power of 10 which is equal' to the given number. But Ihc index of 
any integral power of lO, is one lei's than the number of places in 
that powec, confequently the logarithm of any other number, which 
is no integral power of lo, is not quite one lefs than the number of 
places in that power of the given number whofe index is looo Uc, or 
the logarithm of lo. 

Find therefore the lOth, or looth, or loooth, &c, power of any 
number, as fuppojfe 2, with the number of figures in fuch power 5 then 
fluU that number of figures always exceed the logarithm of a> altho* 
theexcefs will be conftantly lefs than i* 

An example of this procefs is 



here given in the margin ; where 
the I (I column contains the fev- 
tral powers of 2, the 2d their 
correfpondin^ indices, and the 
3d contains the number of pla- 
ces in the powers in the firft 
column ; and of thefe numbers- 
in the third column, fuch as 
are on the lines of thofe indices 
that confift of i with ciphers, 
are continual approximations 
to the logarithm of 2,^berng 
always too great by lefs than 
I in the laft ngure, that logar- 
ithm being 30 1 02999566398 &c. 
And here fince the exaft 
powers of 2 are not required, 
but only the number of figures 
they confift of, as ihewn by 
the third column, only a fc;w 
of the firft figures of the powers 
in the firft column are retained, 
thole being fufficient to deter- 
inine the plumber of places in 
thern^; and the multiplications 
in raifing thefe powers are perJ 
formed in a contracted way, 
fo as to have the fifth or laft 
figure in them true to the near- 
eft unit. Indeed thefe multi- 
plications might be performed 
Jn the fame manner, retaining 
only the firft three figures, and 
thofe to the neaieft unit in the 
third place; wjiich would make 
this a very eafy way indeed of 
finding the logarithms 0/ a few 
l^imt pumbers» 



Fowtr« 

of 2 



t6 

256 



024 
0486 
099s 
2089 



2676 
6069 
5823 
66680 



Indices. 



0715 
1481 
3182 

7377 



9950 
39803 

5843 
25099 



99900 
9980 z 
99601 
99204 



99006 
98023 
96085 

92323 



90498 
81899 
67075 
44990 



36846 

13577 
18433 

33977 



I 

2 

4 
8 



10 
20 
40 
80 



No. ot^'places, 
or logs. 



I 

z 

a 



4 log. of i 
7 log. of 4 
13 log, of 16 
25 log^ of 2j6 



xoo 
260 
400 
800 



1000 
2000 
4000 
8000 



1 0000 
20000 
40000 
80000 



3OH log, 2 

6021 log. 4 
12042 log. 16 
24083 t 256 



I 00000 
200000 
400000 
800000 



I 000000 
2000000 
4000000 
8000000 

I 0000000 
20000000 
40000000 
80000000 



31 log. of a 
61 log.rof 4 
1 21 log. 16 
241 log. 2 j6 



302 log* a 
603 log. 4 

1205 ^^S* '^ 
2409I. 256 



30103 log. 2 

60206 U 4 
120412 $if 
240824 ^ 



301030 
602060 
1 2041 20 
2408240 



3010300 
6020600 
I 2041 200 
24002400 



100000000 
200000000 
400000000 
800000000 



46129 XOOOOOQOOOl 301029996 



30103000 
60206000 

X 2041 1999 

2408^3997 



IC 



tt COI^STRUCTION OP 

It may alfo be remarked, that thofc fcvcral powers, whofe indilsei 
ire I with ciphers, are raifed by thrice fquaring from the former 
powers, and multiplying the firft by the third of thefe fquares ; ma- 
king alfo the correfponding doublings and additions of their indices i 
thus, the fquare of 2 is 4, the fquare of 4 is 16, the fquare of 16 h 
556, and 256 multiplied by 4 is 1024 ; in like manner, the double of 
I is 2, the double of 2 is 4, the double of 4 is 8, and 8 added to 2 
makes 10* And the lame for alt the following powers and indices. 
The numbers in the third column, which (hew how many places 
ftre in the correfponding powers in the firft column, ar6 produced int 
the very iamc way as thofe in the fecond column, namely, by thred 
duplications and one addition ; only obferving to fubtrad i when the 
produd of the firft figures are cxprefled by one figure, c^ when the 
firft figures exceed thofe of the number or power next above them. 
It may farther beobferved that, like as the firft number'in each qua- 
ternion, or fpace of four lines or numbers, in the third column, ap- 
proximates to the logarithm of 2, the firft number in the firft quater-. 
Hion of the firft column ; fo the fecond, third, and fourth terms of 
each quaternion in the third column, approximate to the logarithm of 
4, 169 and 256, the fecond, third, and fourth numbers in the firft 
quaternion of the firft column. And moreover, by cutting off one, 
two, three^ &c, figures, as the index or integral part, from the faid 
logarithms of 2, 4, 16, and 256, the firft, fecond, third and fourth 
numbers in the firft quaternion of the firft column, the remaining 
figures will be the decimal part of the logarithms of the correfponding 
firft, fecond, third, sind fourth numbers in the following fecond, 
tiiird, fourth, &c, quaternions : the reafon of. which is, ti^at any 
number of any quaternion in the firft column, is the tenth power of 
the correfponding term in the next preceding quaternion. So that 
the third column contains the logarithms of all the numbers in the 
firft column : A property which, if Dr. Newton had been aware of, 
he could not well have committed fuch grofs miftakes as are found 
in a table of his fimilar to that above given, in which moft of the 
numbers in the latter quaternions are totally erroneous ; and his con* 
fufed and imperfe^ account of this method, would induce one td 
l|elieve that he did not well underftand it. 

In the fixth chapter our illuftrious author begins to treat of thd 
ether general method of finding the logarithms of prim^ numbers^ 
which he thinks is an eafier way than the former, at Icaft wlien the 
logarithm is required to a great many places of figures. This method 
confifts in taking a great number of continued geometrical meana 
between i and the given number whofe logarithm is required ; that 
is, firft extrading the fquare root of the given number, then the 
TOOt of the firft root, the root of the fecond root, the root of the 
third root, and fo on till the laft root fhall exceed 1 by a very fmall 
decimal, greater or lefs according to the intended number of places 
to be in the logarithm foueht : Then finding the logarithm of this 
fmall number, by methods defcribed below, he doubles it as often as he 
inadc extra^ioAS of the fquare root^ or^ which is the fame thing, he 

multiplier 



LOOARltHMS, 



iDoitipHes It by fuch power of 2 as is denoted by the fald numbe|: of 
^xtradlions, and the refult is the required logarithm of the given 
number ^ as is evident from the nature of logarithms. The rule to 
know how far to continue this extradion of roots is, that the number 
of decimal places in the laft root b^ double the number of true places 
required to be found in the logarithm, and that the firft half of thent 
be ciphers; the integer being i : The reafon of which is, that then 
the fignificant figures in the decimal, after the ciphers, are diredly 
proportional to thofe in the correfponding logarithms $ fuch figures 
m the natural number being the half of thofe in {he next preceding 
number, like as the logarithm of the laft number is the half of the 
preceding logarithm. Therefore, any one fuch fmall number, with 
its logarithm, being once found, by the continual extradions of 
fquare roots out of a given number, as lo, and correfponding bifec«* 
fions of its given logarithm i ; the logarithm for any other fuch fmall 
number, derived by like continual extractions froni another given 
number, whofe logarithm is fought, will be found by one fingle pro- 
portion ; which logarithm is then to be doubled according to the 
number of extractions, or multiplied at once by the like power of 2^ 

for the logarithm of the number propofed. 
To find the firft fmall number and its 
logarithm, our author begins with the 
number 10 and its logarithm i, and ex- 
tracts continually the root of the laft 
number, and bifeCts its logarithm, as 
here regiftered in the margin, but to far ^^' ^^* t 



I 

2 

3 

4 

i 



10, given no. limits io^. 



3*162277 &c 
1778:79 
1-333521 
1-154781 
1-074607 
&c, 



o-s 
0-25 

0-I25 

0-0625 
0*03125 



more places of figures, till he arrives at the 5^d and 54th roots^ with 
their annexed logarithms, as here below ; 

Numbers. I Logarithms. 

53 fOOOOO,OOOOr,00000,25 563,82986^0064,70 Io*00000,00000,60000, 1 1 IOZ,23024,625I5|65jj()^ 

^4 i'Ooooo,otocc,ococc,i278i,9i493,zoo32,35lo*ooooo,ooooo,ooooo,0555i|i 1512,31257,8*705 

where the decimals in the natural numbers are to each other In the 
ratio of the logarithms, namely in the ratio of 2 to i : and therefore 
any other fuch fmall number being found, by continual extraction or 
otherwife. it will then be as 12781 &c, is to 5551 &c, fo is that other 
fmall decimal, to the correfponding fignificant figures of Its logar- 
ithm. But as every repetition of this proportion requires both z 
very long mi|ltiplication and divifion, he reduces this conftant ratio 
to another eq|.iivalent ratio whofe antecedent is i, by which all the 
divifions are faved : thus, 

as 12781 &c ; 5551 &c : : looo &c : 43429448190325 1 804, 
that is, the logarithm of i'ooooo,ooooo,ocx)00,i 

is 0*00000500000,00000, 04342,9448 1, 90325, 1804 ; 
and therefore this laft number being multiplied by any fuch fmall 
decimal, found as above by continual extraction, the product will 
be the correfponding logarithm of fuch laft root. 

But as the extraction of fo many roots is a very troublefome ope** 
fjtion^ oifr autbcjr devifes fome ingenious contrivances to abridge that 

|sibou^ 



H 



CONSTRUCTION OF 



labour. And firft, in the 7th chapter, by the following device, to 
have fewer and eafier extradlions to perform : namely^ raifing the 
powers from any given prime number, whole logarithm is fought, 
till a power of it be found fuch that its firft figure on the left hand is 
I, and the next to it either one or more ciphers ; then, having di- 
vided this power by i with as many ciphers as it has figures after the 
firft, or fuppofing all after the firft to b^ decimals, the continual 
roots from this power are extracted till the decimal become fufficiently 
fmall, as when the firft fifteen places are ciphers ; and then by mul- 
tiplying the decimal by 43429 &c, we have the logarithm of this laft 
root^ which logarithm multiplied by the like power of the number 
S{, gives the logarithm of the firft number fronj which tne extrac* 
tion was begun : to this logarithm prefixing a i, or 2, or 3, &c, ac- 
cording as this number was foiind by dividing the power of the given 
prime number by 10, or 100, or 1000, fcc; and laftly, dividing the 
rcfult by the inqex of that power, the quotient will be the required 
Jogarithn) of the given prime number. Thus, to find the 
logarithm of 2 : it is firft raifed to the lotb power, as in 
(he margin, before the firft figures come to be 10 ; then, di- 
viding by 1000, or cutting off for decimals all the figures 
after the firft or i, the root is continually extrafted, from the 
quotient 1,024, **'' ^be 47th extraftion, which gives 
1,00000,00000,00000,16851,60570,53949,77 5 thedecimal 
part of which multiplied by 43429 &c, gives > 
0»00000,ooooo,ooooo,073 1 8,55936,90623,9368 for its log- 
arithm : and this being continually doubled f or 47 times, 
will give the logarithms of all the roots up to 
the firft number : or being at once multiplied 
by the 4ythpower of 2, viz. 140737488355328, 
which 18 raifed as in the margin, it gives 
0,01029499566)3981 1,95265,27744 for the 
logarithm of the number 1,024, true to 17 or 
1 8 decimals : to this prefix 3, fo {hall 3,0102 
^c be the« logarithm of 1024 : and laftly, be- 
caufe 2 is the tenth root of 1024, divide by 10, 
fo fliall 0,30102,99956,63981,1952 be the 
logarithm required to the given number 2* 
^he logarithms of i, 2, and 10 being now 

known; it is remarked that the logarithm ,.^...._ 

of 5 becomes known ; for fince 104-2 is = 5, therefore log. 10 — - 
log. 2 =: log. 5, which is 0,69897,00043,36018,8058 ; and that from 
the multiplications and diviuons of thefe three 2,5,10, with the cor- 
refponding additions and fubtradions of their logarithms, a multitude 
of other numbers and their logarithms are produced ; fo from the 
powers of 2 are obtained 4, 8, 16, 32, 64, &c; from the powers of 5 
thefe 25, 125, 625, 3125, &c ; alfo the powers of 5^ by thofe of. 10 
give 250, 1250, 6250, &c ; and the powers of 2 by thofe of 10 give 
20, 200, 2000, &c ; 40, 400, 80, 800, &c J likcwifc by divifion arc 
obtained af, i|, 12:, 6^, ||, 3t, 6j, &c, 

He 



2 


I 


- 4 
8 

16 


2 

3 

4 


3^ 
64 


I 


128 


7 


a 56 


8 


S^^ 


9^ 


1024)101 



4 
8 

16 

32 

64 

128 

256 

512 

1024 

• 1048576 

1073741824 

10995 1 1627776 

1407374883553^8 



3 

4 

5 

6 

7 
8 

9 

10 

40 

30 
401 
47 



LOG A R I TH M S. 6^ 

He then obferves that the logarithm of 3, the next prime nutitber, 
will be! beft derived from that of 6, in this manner : 6 raifed to the 
9th power becomes icx>77696, which divided by looooooc, gived 
1,0077696, and the root from this continually extra£ted till the 46th9 
is 1,00000,00000,00000, 10998,59343,88 155,7 1 866 i ^^ decimal 
part of which multiplied by 43429 &:c, gives 
0,00000,00000,00000,04776,62844,78608,0304 for its logarithm; 
and this 46 times doubled, or multiplied by the 46th power of 2, gives 
0,00336,12534,52792,69 for the logarithm of 1,0077696 ; to which 
adding 7, the logarithm of the divifor loooooco, and dividing by 
9, the iiidexof the power of 6, there refults 0,77815,12503,83643,65 
zor the logarithm of 6 ; from which fubtradling the logarithm of 2> 
there remains 0,47712^12547,19662,4.4 for the logarithm of 3. 

In the eighth chapter our ingenious author defcribes an original 
and eafy itiethod of conftru£ling, by means of differences, the con- 
tinual mean proportionals which were before found by the extraction 
of roots. Ana this, with the other methods of generating logarithms 
by differences, in this book as well as in our author's Trigonometria 
Brifaunicaj are I believe the firft inilances that are to be found of 
making fuch uie of differences, and fhew him to have been the in* 
ventor of what may be called the DiJTerential Method. He feems to 
have dilcovered this method in the following manner : Having ob- 
fervcrd that thefe continual means between i and any number propo- 
fed, found by the continual extradion of roots, approach always 
nearer and nearer to the halves of each preceding root, as is yifible 
when they are placed together under each other ; and indeed it ii 
found that as many of the fignificant figures of each decimal part, as 
there are ciphers between them and the integer i, agree with the half 
of thofe above them ; I fav, having obierved this evident approxi- • 
mation, he fubtraded each of thefe decimal parts, which he called 
A or the firft differences, from half the next preceding one, and by 
comparing together the remainders or fecond differences, called B^ 
jbe /ound that the fucceeding were always nearly equal to i of the 
Aext preceding ones ; then taking the difference between each fecond 
difference and i of the preceding one, he found that thefe third dif« 
iterences, called C, were nearly in the continual ratio of 8 to i ; again 
taking the difference between each C ai^d f of the next preceding, he 
found that thefe fourth differences, called D, were nearly in the con- 
tinual ratio of 16 to i j and fo on, the 5th (E), 6th (F), &c, diffe- 
svpcesy b^g nearly in the continual ratio of 32 to i, of 64 to 1, &c : 

K * thefe 



•46 



CONSTRUCTION OF 



thefe plain obfervatiofis be- 
ing m^adc, they very natu- 
rally and clearly fuggefted 
to him the notion and 
method of conftruding all 
the remaining numbers 
firom the differences of a 
few of the firft, found by 
extra£ting the roots in the 
ufual way. This will evi- 
dently appear from the 
annexed ft)ecinaen oPa few 
of the (irft numbers in the 
laft example for finding 
the logarithm of 6 ; where 
after the 9th number the 
reft arc fuppofed to be con- 
AruAed from the precedmg 
difFerences of each, as here 
ihewn in the loth and 
iith. And it is evident 
that, in proceeding, the 
trouble will become al- 
-ways lefs and lefs, the dif- 
ferences gradually vanifh- 
iogy till at lafl only the 
firft difFerences remain. 
And that generally each 
lefs difference is fhorter 
than the next greater, by 
as many places as there are 
ciphers at the beginning of 
the decimal in the number 
to be generated from the 
differences. 

He then concludes this 
chapter with an ingenious^ 
but not obvious, method 
of finding the differences 
B^ C,D, E, &c, belonging 
tjsLan/ number, as fuppoie 
'W|t*-9th9 ftom that number 
itfelf, i^d^endent of any 
of the preceding 8th, 7th; 
6th, 5th, &c; and it is 
this: Raife the decimal 
thenjwill the 2d (B}^ 3d 
bel6v^viz. 



X 

f 

3 

4 



■ I 1 ■ ■ — — ■— ^w— 

1,00776,96 

1,00387,71833. 369$i,45663,S46s5, 1 

i,X)Oi93,6766i,36946,6i67«;,S702i,9' 

1,00096,79146,39099,01728,89072,0 

1,0004s, 38402,68846,6198 s>49^Sl»5 .. 



10 



1,00024,18908,78824,68563,80872,7 
24,19101,34423,31492,74626,7 

»9^>5 5 5')8^ i9^<)3754-o 



1,00012,0938 1,»63 ,,7, 1 34<;9,439i9>4 

i2p94S4,394U, 3428 1,90436,3 

73,13015.20822,46516,9 

73ii3899/>5732>i3438»5 
884,44909,7692 1, S 

1, 50006,04672, 3 50^ 5,30968,01600,5 

6,04690,63 1 98,56719,7 I959i7 

.18,28143,25761,70359,2 

18,282^3,80205,61629,2 

110,54443,91270,0 

110,55613,72115,4 

Tf6Q i^p^4^,2 



I 



i,oooo3,.:ij3 1, (>05C 5, 65775,90479,4 

3,02336, [7527,65484,001500,2 

4,257021,99708,04320,8 

4l57035lS^44o^4•*5''<9»8 
^181731.3^^69,0 

13,81805.43908,7 

73,10639,7 

73,11302,8 

663,1 



1,00001,5x164,6^999, 05672,95049,8 
1,51165,80252,82887,98239,7 
1,14253,77215,03190,9 
Hitherto the i,i4255,i<9927,oio8oyX 
fmallcr differences 1,7271 1,97889,3 * 
are found by fub- 1,72716,54783,6 
trading Che larger from V/5 6894,3 
the parts of the like pre- 4,5691 5,0 
ceding ones. 20,7 

20,7 



Here the greater diJfcrenccs 65" 

remain after fubtradting 28555,89 

the fmaUer from Che parts 28555,24 

of the difference of 21588,99736,16 

the next preceding 21588,71x80,92 

number. 28563,44303,75797,72 

28563,22715/34616,80 

75582,32999^52836,47524,40 

r,co-oo,755S2,o4436,3ri2 1,42907,60 



IT 



2 

'78-^,70 

1784,68 

2698,58897,62 

2698,57112,94 

7140,80678,76154,20 

7140.77980,19041,26 

37791,02218,15060,71453,80 

1,000^0,37790,95077,37080,5241 2,54 



A 
B 

iA 
B 

JB 
C 



A 
JA 

K 
JB 

C 

% 

A 

iA 

B 

iB 
C 

ic 

D 
E 



A 

B 

iB 
C 

D 



iA 
A 



' F 

^n 
c 

IB 
B 

lA 

A 



A to the 2d, 3d, 4th, 5th, &c powers 5 
(C), 4th (D), &c differences, be as here 



LOGARITHMS. 

B=|A% 

C= . lA' + JA*, 

iAHiA'+T\A»+ |A'+ V*A% 

. 33A5+ 7A*+ioViA'+ iJtVtAM- naA»+ rlUA", 

. I3t''jA*+ 8.iA'+ 296 ;VA'+ 834.V,A»+ i9S3?llA"&c. 
. . , iaJx',A'+isio/^TA'+i'475-i'fVA»+ bS^J'.iir^"! kc. 

. 1937^VA'+47iS'tVtA»+ 70684$ J«iA"&c. 
. S4902tV,A»+jss846s1J5-*JA'?8cc. 

'•a8o?S27A*&c, 



^ • 



Thus in the 9th number of the foregoing example, omitting" th« 
ciphers at the beginning of the decimals, wc have. 
A =1,51164,65999,05672,95048,8 
A* = - 3,2850754430,06381,6726 
A^ = - - - 3,454^^»65^ 39*48546,2 

A+*= 5,22156,97802,288 

A5 = 7,89316,8205 

A* = . - 11,93168,1 

&c. 

Confequently 

-JA*=* i,i4253»77"5>03'^9o»8363 = B 

|A3 - i,727J">326i9»74273 ' ^ 

|A* - - - 6526962225 

|A' + |A^ 1,7271., 97889,36498 =C 

IA\ 4,56887,.^5577 
I A* - - 6,90652 

f A^+i A^ +t'8A* 4,56894>^6234 = D 

2|A5 - . 20,71957 
7A* - ' ■ 83 .^ 
2|A* + 7A* - . - 20,72040 zr E 
which agree with the like differences in the foregoing fpecimen. 

In the ninth chapter, after obferving that from the logarithms of 
I, 2» 39 5> and 10, before found, are to be determined^ by addition 
and fubtradion, the logarithms of all other numbers which can bje 
produced from thefe by multiplication and divifion; for findAg the 
logarithms of other prime numbers, inflead of that in the feventh 
chapter, our author then ihews another ingenious method of ob« 
taining numbers beginning with i and ciphers, and fuch as to bear 
a certain relation to fome prime number by means of which^its log- 
arithm may be found. The method is this : Find three produ<Els hal- 
ving the common difference i, and fuch that two of them are produ- 
ced from fa6lors having given logarithms, and the third produced 
from the prime number, whofe logarithm is required, either multi- 
t'iplled by itfelf. or by fome other number whofe logarithm is g\vetfi 

K a then 



69 CONSTRUCTION pP 

then the greateft and leaft of thefe three products being multiplied 
together, and the mean by itfelf, there arife two other produds alfo 
differing by i, of which the greater divided by the lefs, gives for a 
quotient i with a fmall decimal, having feveral ciphers at the begin- 
ning. Then the logarithm of this quotient being found as before, 
from thence will be deduced the required logarithm of the given 
prime number. Thus, if it be propofed to find the logarithm of the 
prime number 7 ; here 6x8 = 48, 7x7= 49, and 5 x 10 =: 50 
will be the three produfls, of which the logarithms of 48 and 50, 
the I ft and 3d, will be given from thofe of their fzStors 6, '8, 5, 10 5 
9lfo 48 X 50 = 2400, and 49 x 49 = 2401 are the two new produdls, 
and 2401 -T 2400 = 1000417 their quotient : then the leaft of 44 
,means between i and this quotient is 1,00000,00000,00000,02367, 
98249,04333,6405, which multiplied by 43429 &c, produces 
0,00000,60000,00000,01028,40172,88387,29715 for its logarithm ; 
which being 44 times doubled, or multiplied by 17 592 1 860444 16, 
pr6duces 0,00018,09183,45421,30 for the logarithm of the quotient 
1, 00041 f ; which being added to the logarithm of the divifor 2400, 
gives the logarithm of the dividend 2401 ; then the half of this log- 
arithm is the logarithm of 49 the root of 2401, and the half of this 
again gives 0,84509,80400,14256,82 for the logarithm of 7 which is 
the root of 40.<— The author adds another example toilluftratetbis 
method^ ana then fets down the requifite fadors, products, and quo- 
tients for finding the logarithms of all other prime numbers up to 
100. 

The loth chapter is employed in teaching how to find the logar- 
ithms of fradions, namely by fubtra^ling the logarithm of the deno- 
minator from that of the numerator, then the logarithm of the frac- 
tion is the remainder ; which therefore is either abundant or defec- 
tive, that is pofitive or negative, as the fradion is greater or lefs 
than I. 

In the nth chapter we are (hewn an ingenious contrivance for very 
accurately finding intermediate numbers to given logarithms, by the 
proportional parts. On this occafion it is remarked, that while the 
abfolute numbers increafe uniformly, the logarithms increafe unequally^ 
with a decreafing increment ; for which reafon it happens, that either 
logar^ithms or numbers correded by means of the proportional parts, 
will not l^e quite accurate, the logarithms fo found being always too 
fn:iall, and the abfolute numbers fo found too great ; but yet fo how- 
ever as that they approach much nearer to accuracy towards the end 
of the table, where the increments or differences become much 
nearer to equality, than in the former parts of the table. And from 
this property our author, ever fruitful in happy expedients to obviate 
natural difficulties, contrives a device to throw the proportional part, 
to be found from the numbers and logarithms, always near the end of 
the table, in whatever part thev may happen naturally to fall. And 
it is this : Rejecting the chara^eriftic of any given logarithm, whofe 
number is propofed to be found, take the arithmetical complement of 
the decimal part, by fubtrafling it from 1,000 &c, the logarithm of 
of IP i then find in the tablo the logarithm next leis than this arith- 
metical 



Y.06ARITHMS. 69 

metical complement, together with its abfolute iiumber; to this ta- 
bular logarithm add the logarithm that was given, and the fum will 
be a logarithm neceflarily falling among thofe near the end of the ta-*' 
ble ; find then its abfolute number, correded by means of the pro* 
portional part, which will not be very inaccurate, as falling near the 
end of the table ; this being divided by the abfolute number, before 
found for the lozarithm next lefs thaq the arithmetical complement, 
the quotient will be the required nuniber anfwering to the given log- 
arithm ; which will be much more correct than if it had been found 
from the proportional part of the difference where it naturally hap- 
pened to fall : and the reafon of this operation is evident from the 
nature of logarithms. But as this divtlbr, when taken as the num- 
ber anfwering to the logarithm next lefs than the arithmetical com* 
plement, may happen to be a large prime nunjber ; it is farther re- 
marked, that inftead of this number and its logarithm, we may uf^ 
the next lefi compofite number which has fmalT factors, apd its log- 
arithm ; beca\ife the divifion by thofe fmall fad^ors, Inftead of by the 
number itfelf, will be performed by the fhort and eafy way of divi- 
fion in one line. And for the more eafy finding proper compofite 
numbers and their favors, oiur author here fubjoins an abacus or lift 
of all fuch numbers, with their logarithms and component fadors, 
from loco to lOOOO ; from which the proper logarithms and fa6lors 
are immediately obtained by infpcdlion. Thus, for example^ to find 
the ropt of loSoQ, or the mean proportional between i and 10800 : 
The logarithm of 10800 is 4903342,37554,8695, the half of which is 
2,01671,18777,4347 the logarithm of the number fought, the arith« 
metical complement of which logarithm is 0,98328,81222,5653; novr 
the neareft logarithm to this in the abacus is 0,98227,12330,3957, 
and its annexed number is 9600, the factors of which are 2, 6, 8 ; to 
this lafl logarithm adding the logarithm of the number fought, the 
fum is 0,99898,31107,8304) whofe abfolute number, correded by 
the proportional part, is 99766,12651,6521, which being divided 
continually by 2, 6, 89 the fadors of 96, the lafl quotient is 
103,92304845471; which is pretty correct, the true number bdng 
103,92304.8454133=^^10800. 

We now arrive at the 12th and 13th chapters, in which our inge- 
nious author ftrfl of all teaches the rules of the Diffierential Mithod^ 
in conftruSing logarithms by interpolation from differences. This 
is the fame method which has fince been more largely treated of by 
later authors, and particularly by the learned Mr. Cotes in his Cano^ 
notechnia. How Mr. Briggs came by it, does not well appear, as he 
only delivers the rules, without laying down the principles or in» 
veftigation of them. He divides the method into two cafes, namely- 
when the fecond differences are equal or nearly equal, and when the 
differences run out to any length whatever. The former of thefe is 
treated in the 12th chapter; and he particularly adapts it to the in- 
terpolating 9 equidiftant means between two given terms, evidently 
for this reafon, that then the powers of 10 become the principal mul- 
tipliers or divifors, and fo the operations performed mentally. The 
fubftance of his procefs is this ; Having given two abfolute numbers 

with 



70 



CONSTRUCTION OF 



with their logarithms, to find the logarithms of 9 arithmetical means 
between the given numbers : Between the given logarithms take the 

I ft difference, as well as between each of them 
and their next or equidiftant greater and Jefs log- 
arithms ; and likewife the 2d differences, or the 
two differences of thefc three ift differences ; then 
if thefe 2d differences be equal, multiply one of 
them feverally by the numbers 45, 35, &c, in the 
annexed tablet, dividing each produ£t by 1000, 
that is cutting off three figures from each ; laftly 
to 7^ of the ift difference of the given logarithms 
add feverally the firft'five quotients, and fubcra<fl 
the other five, fo fliall the ten refults be the refpeflive ift differences 
to be continually added, to compofe the required feries of logarithms. 
Now this amounts to the fame thing as what is at this day taught in 
the like cafe : we know that if -^be any term of an equidiftant feries 
of terms, and ^, ^, r, &c, the firft of the ift, 2d, 3d, &c, o jer of 
differences ; then the term z, whofe diftancc from A is exprcn- j by 

*, will be thus, z=:y/4"*tf + x. h + Jf. . c &c. And 

if now, with our author, we make the 2d differences equal, then r, rf, 
iy &:c, will all vanifh or be equal to o, and z will become barely 



1 


4f 


' 5? « 


2 


3' 


; .tec 


3 
4 


2( 

II 




5 






6 




' l^ 


7 


I( 


; 'cgcc 


8 


2( 


> 3 a, 


9 


3! 


10 


4! 


. w 



Therefore if we take a- 
fucceifively equal to 
WTO* WT5, &c, we fliall 
have the annexed feries of 
terms with their differen- 
ces. Where it is to he 
obferved, that our author 
had reduced the differen- 
ces from the ift to the 2d 



Series of terms. 
A 
A\^a\^l I 



^+tV + i 






A^r^\a\-i^^h 



The Differences* 









= tV+ 



— -v 






iV^TToTc^ 









i3* 



To^ 



- ^.^ a i I 






form, as he thought it eafier to multiply by 5 than to divide by 2« 



A — I 



Alfo all the laft terms ^a-. h) are fet down pofitive, becaufe in 

2 

the logarithms h is negative. — «— If the two 2d differences be only 
nearly equal, take an arithmetical mean between them, and proceed 
with it the fame as above with one of the equal 2d differences.-rHe 
alfo fliews how to find any one fingle term, independent of the 
reft ; and concludes the chapter with pointing out a method of 
finding the proportional part more accurately than before. 

In the 13th chapter our author remarks, that the beft way of filling 
up the intermediate chiliads of his table, namely from 20000 to 90000, 
is by quinquifeftion, or interpofing four equidiftant means between 
two given terms ; the method of performing which he thus particu^ 
larly defcribes. Of the given terms, or logarithms, and two or three 
Others qh each -fide of them, take the ift;^ 2d, 3d, &c differences. 

m 



LOGARITHMS. jt 

till the laft differences come out equal, which fuppofe to be the 5th 
differences: divide the ift differences by 5, the ad by 25, the 3d bjr 
125, the 4th by 625, and the 5th by 3125, and call the refpefkive 
quotients the id, 2d, 3d, 4th, 5th mian differences; or» inftead of 
dividing by thefe powers of 5, multiply by their reciprocals -Z^^, yl^, 
Wttsj Wgscj T-siioz* that is multiplied by 2, 4, 8, 16, 32, cutting 
off refpe<3ively one, two, thr^e, four, five figures from the end of the 
products, for the feveral mean differences : then the 4th and 5th of 
thefe mean differences arc fufficiently accurate, but the ift, ad, and 
3d are to be correded in this manner ; from the mean third differen- 
ces fubtra£t three times the 5th difference, and the remainders are the 
correQ 3d differences ; from the mean 2d differences fubtrad double 
the 4th differences, and the remainders are the correS 2d differences; 
laftly from the mean ift differences take the correft 3d differences, 
and -^ of the 5th difference, and the remainders will be the coittSt 
firft differences. Such are the corrections when the differences extend 
as far as the 5th. However in compleating thofe chiliads in this 
way, there will be only 3 orders of differences, as neither the 4th nor 
5th will enter the calculation, but will vanifli through their fmallnefs : 
therefore the mean 2d and 3d difference will need no correction, and 
the mean firft differences will be correfled by barely fubtrading the 
3d from them. Thefe preparatory numl)ers being thus found, all 
the 2d differences of the logarithms required, will be generated by 
adding continually, from the lefs to the greater, the conftant 3d 
difference ; and the feries of ift dii&rences will be found by adding 
the feveral 2d differences; and laftly by adding continually thefe ift 
differences to the ifi given logarithm &c, the required logarithmic 
terms will be generated. 

Thefe eafy rules being laid down, Mr. Brlggs next teaches how 
by them the remaining chiliads may beft be compleated : namely, 
having here the logarithm for all numbers up to 20000, find the log- 
arithm to every 5 beyond this, or of 20005, 20010, 20015, &c, ia 
this manner ; to the logarithms of the 5ih part of each of thofe, 
namely 4001, 4002^ 4003, &c, add the conftant logarithm of 5, and 
the fums will be the logarithms of all the terms of the feries 20005, 
20010,20015, &c : And thefe logarithms will have the very fame dif- 
ferences as thofe of the feries 4001,4002,4003, &c ; by means 
of which therefore interpofe 4 equidlftant terms by the rules above) 
and thus the whole canon will be eafily compleated. 

He here alfo extends the rules for correfting the mean differences 
in quinquife&ion, as far as the 20th difference ; he alfo lays down 
fimilar rules for trifedlion, and fpeaks of general rules for any other 
iedion, but omitted as being lefs eafy. So that he appears to have 
been poffeffed of all that Cotes gfterwards delivered in his CanO'^ 
notechnia Jive ConJiruRio Tabular urn per Differ entiasy drawn from 
the Differential Method^ as their general rules exadlly agree, Briggs's 
mean and correit differences being by Cotes called round and 
quadrat differences, becauie he expreffes them by the numbers i, 2^ 
3* &c. written refpedtively in a fmall circle and fquare. 

Mr, 



^i CONSTRUCTION OF 

Mr. Btigg$ alfo obferves that the fame rules equally ,ap{$1y to tite 
conftru6lion of equtdiftant terms of any other kind, fuch as fines, 
tangents, fecants, the powers of numbers, &c: and farther reniarks, that 
of tne fines of three eqiiidifferent arcs, all the remote differences li^y 
be found by the rule of propbrtion, becaufe the fines and their 2d, 401, 
6th, 8th, &c differences' are continued proportionals, as are alfo the 
ift, 3d, 5th, 7th, &c differences among themfelves ; and like as the 
id, 4th^ 6th, &c differences are proportional to the fines of the mean 
arcs, fd alfo are the ifl, 3d, 5th, &c differences proportional to the 
cofines of the fame arcs. Moreover with regard to the powers of 
numbers, he remarks the following curious properties; ift, that 
they will each have as many orders of differences as are denoted by 
the index of the power, the fquares having two orders of differences, 
the cubes three, the 4th powers four, &c : fecond, that the lafl dif- 
ferences will be all equal, and each equafl to the common difference 
of the fides or roots raifed to th6 given power and multiplied by 
1x^x3x4 &c, continued to as many terms as there are units In 
the ihd*x ; fo if the roots differ by i, the 2d difference of the fquares 
wrfl bt each i x 2 or 2, the 3d differences of the cubes each 1x2 
X 3 ot 6, the 4th differences of the 4th powers each 1x2x3x4 
^24, and fo on ; and if the common difference of the roots be any 
other number n, then the lafl differences of the fquares, cubes, 4th 
jtowers, 5th powers, &c, will be rcfpeSively an*, 6«3, 24^4, i20»5, 
&c. 

Befides what was fliewn in the eleventh cliapter concerning the 
fating out the logarithms of large numbers by means of proportional 
parts, he employs the next or 14th chapter in teaching how, froni 
* thefirfl ten chiliads only, and a fmall table of one page, here given, 
to find the number anfwering to atiy logarithm, and the logarithm to 
any nirrtber confifting of fourteen places of figures. • 

Having thu$ fully (hewn the conflrudioil and chief properties of 
his logarithms, our ingenious author, in the remaining eighteen 
chapters, exemplifies their ufes in various curious and important 
fubjeSls; fuch as The Rule-of- three, or rule of proportion j finding 

^— ^— — — — ■ ■ I ■ I II i. I . ■ m a m • • . ■ ■■ , , 

* It is no more than a large exemplification of this method of Briggs's that has hettK 
printed fo late as 177 1, in a 4x0. tra^ by ^i*. Rob. Flower^ under the title of The Radix, 
^ m New /f^ of making Logarithms, Although Briggs's worlT might not be known to this 
writer. — ^Since this was written 1 have been favoured with the following anecdote, con- 
cerning Mr. Flower and his work, by the Rev. Dr. Horfley, the learned editor of tho 
works of Sir I. Newton. '< This Robert Flower was a very obfcure, and probably an 
illiterate man. He was mafler of a wrifiiig fchool in the town of Bifhop Stortford in 
Hertfbrdfhire. He communicated' liis Radix, before he publifhed it, to my late learned 
friend Math. Raper, Efqt of Thorley Hall. I was at Thorley at the time, upon a 
vifit to my father, who was re^or of th6 parifh ; and I well reraejnber that ^Ir. R»per 
told me With great furprize, that FloU'er (who was known to us both by name as the 
irriting-mafter of the neighbouring market town) had fallen upon Briggs'sway of firid^' 
ing all logarithihs from the iirlt ten chiliads. And he was fo well perfuaded that 
Flower had made the difcoyery for himfelf, without any light from Briggs, that with 
his aCcuftomed munificence h6 rewarded the man's ingenuity with a prefent of tea* 
guineas { informing him I believe that his work had been done Worei ana difluading thv 
publication*" 



L O d A tt 1 1 tt M S* .7j 

•the H:>ot5 of given numbers } finding any number ofme^ propoiw 
'tionals between two given terms; with other arithmetical rules: 
Alfo various geometrical fubjedts, as ift. Having given the fidvs of 
any plane triangle^ to find the area, perpendicular, angles, and dia« 
meters of the infcribed and circumfcribed circles ; 2d, In a right- 
angled triangle, having given any two of thefe, to find the reft, viz. 
one leg and the hypotenufe, one leg and the fum or difference of the 
•hypotenufe and the other leg, the two legs, one leg and the area, the 
area and the fum or difference of the legs, the hypotenufe and fum 
or difference of the legs, the hypotenufe and area, and the perimeter 
and area ; 3d, Upon a giVen bafe to defcribe a triangle equal an4 
ifbperimetrical, to another triangle given 5 4th, To defcribe the cir- 
tinhference of a circle fo, that the three didances from any point in 
it to the three angles of a given plane triangle, (hall be to one ano- 
ther in a given ratio ; 5th, Having given the bafe, the area^ and tht 
I'atio of the two fides of a plane triangle, to find the fides j 6tht 
Given the bafe, difference of the fides, and area of a triangle, tofin^ 
the fides ; 7th, To find a triangle wh6k area and perimeter ihalt be 
expreffed by the fame number ; 8th, Of four given lines, of which 
the fum of any three i^ greater than the fourth, to form a quadrila- 
teral figure about which a circle may bedefcribed ; 9th, Of the dia* 
tnetef, circumference, and area of a circle, and the furface and foli^ 
dity of the fphete generated by it, having anyone given, to find any 
of the reft; !iOth, Concerning the elHpfe, fpheroid, and gauging; 
iith. To cut a line or a number in extreme and mea& ratio; 12th, 
Given the diameter of a circlei to find the fides and areas of the hi*- 
fcribed and circumfcribed regular figures of 3, 4, 5, 6, 8^ io, 12, and 
fifteen fides ; 13th, Concerning the regular figures of 7, 9, 15, 24^ 
and 3ofides'; 14th, Of ifoperimetrical regular figures; 15th, Of equal 
regular figures ; and i6th. Of the fphere and "the 5 regular bodies ; 
Winch clofes this introdudion. Such of thefe problems as can ad« 
toiit of it, are determined by elegant geometrical conftrufltons, and 
they are all illuftrated by accurate arithmetical calculations per-* 
formed by logarithms ; for the exemplification of which they are 
purpofely given. 

At the end he remarks, that the ch'tef and moft neceflary ufe of 
logarithms, istn thedodrineof fphcrical trigonometry, which he herd 
promtfes to give in a future wOrk, and which was accomplifhed in 
his Trigontmetria Britanni<a^ to the defcription of which Vf^ now 
proceed. 

« 

Of B-RIGGS'/ Trig^nmetria Britannica. 

At the clofe pf the account of writings on the natural fines, tan<« 
gents, and fecants, I omitted the defcription of this work of our 
learned author, although it is perhaps the greateft of this kind, all 
things confidered, that ever was executed by one perfon; purpofely 
le&rving my account of it to this place, not eolj^^ it is conne^ed 
with ^ invflAtion md CMftrn^tioA of logarithms^ but thiojkin^ it 

]^ defexved 



74 CQNSTRUCTION OF 

-deferred more peculiar and didinguiihed notice, on accomit of tM 
Importance and originality of its contents. The diVifion of the 
quadrant, and the mode of conilruction,are both new ; and the num- 
bers are far more accurate, and are extended to more places, than 
4hey had ever- been before. The circular arcs bad always been di« 
vided in a fexagefimal proportion ; but here the quadrant is divided 
into degrees and decimals, as this is a much eafier mode of compu- 
tatioathan by 6oths; the diviiion being compleatcd only to icx)thsof 
degrees, though his deiign was to have extended it to loooths of de- 
•grees. And, beiides his own private opinion, he was induced to 
. adopt this niethod of decimal divifions, partly at therequeft of other 
perfons, and partly perhaps from the authority of Vieta, pa. 29 Ca- 
Undarii Gregorianim And it is probable that computations by this de- 
cimal divifion would have come into general ufe, had it not beeil for 
the publication of Vlacq's tables, which were extended to every 10 
feconds, or 6th parts of minutes. But befules this method by a de* 
cimal divifion of the degrees, of which the whole circle contains 
360, or the quadrant 90, in the 14th chapter he remarks that fome 
other perfons were inclined rather to adopt a compleat decimal divi- 
iion of the whole circle, firll into 100 parts,' and each oC thefe intQ 
1060 parts ; and for their fakes he ilibjoins a fmall table of the fines 
of every 40th part of the quadrant, and remarks that from thefe few 
Ihe whole may be made out by continual quinquife£lions } namely, 
5 times thefe 40 make sioo, then 5 tifncs thefe give 1000, thirdly* 
5 times thefe give 5000, and laflly 5 times thefe give 2500P for the 
whole quadrant, or looooo for the whole circumference. 

But to return. Our author's large table confifts of natural fines 
1015 places, natural tangents-and fecants'each to 10 places, logarith- 
mic fines to 14 places, and logarithmic tangents to 10 places, each 
befides the charaiSeriftic. A moft f^upendous performance ! The 
table is preceded by an introduction, divided into two books, the 
one containing an account of the truly ingenious conftrudion of the 
table, by the author himfelf ^ and the other its ufes in trigonometry, 
&e, by Henry Gellibrand, profeflbrof aftronomy in Grefham College, 
who remarks in the preface that the work was compofed by the au- 
thor about the year j6oo ^ though it was only pubUfiied by the di- 
ledion of Gelliorand in 1633, it having been printed at Gouda un« 
der the care of Vlacq, and by the printer of his TrigoMmetria Artifi^ 
gialisf which came out the fame year. 

After briefly mentioning the common methods of dividing the 

3uadrant, and conftru&ing the tables of fines, &c, from the ancientSv 
own to his own. time, he haftens to the defcription of his own jpe^ 
culiar and truly ingenious method, which i& briefly this: having nrft 
divided the quadrant into a fmall number of parts, as 77, be finds the 
fine of one of thofe parts, then /rom it the fines of the double, triple, 
^luadruple, &c, up to the quadrant or 72 parts* He next quJnqui- 
fedseachpf thefe parts, by interpofing four equidiftant means, bydif* 
ferences; he then quinquifecSs each of thefe; and finally eachof chefe 
•agsins which compleats the .divifion ^ Ux^ degrees And centeiins* 



LOGARITHMS^ 



H 



7'fee rules for performing all thefe things, be inveftigates and ilhi'^ 
trates in a very ample manner. In treating of multiple and fubmul^ 
tiple arcs, he gives general algebraical expreffions, for the fine or 
chord of any ^multiple whatever of a' given arc, which be deduce^ 
from a geometrical figure, by finding the law for the feries of fuccef- 
five multiple chords or fines, after the manner of Vieta, who was the 
firft perfon that 1 know of, who laid down general rules for the choixls 
of multiple and fubmyltiples of arcs or angles: and the fa'nle was 
afterwards improved by Sir I. Newton, to fuch form, that radius, 
and double the cofine of the firft given angle, are the firft and lecond 
terms of all the proportions for finding the fines and cofines'of the 
multiple angles. For affigning the coefficients of the terms in the 
multiple expreffions, our author here delivers th? conftru^lion of figu* 
rate or polygonal numbers, inferts a large table of them, and 
teaches their feveral lifes ; one of which is that every other number 
taken in the diagonal lines, furniflies thexoefficients of the terms of 
the general equation by which the fines and chords of multiple arcs 
areexprefled, which he amply illuftrates ; and another, that the fame 
diagonal numbers confiitute the coefficients of the terms of any power 
of a binomial ; which property was alfo mentioned by Vieta in his jjfn^ 
gulares Se^liones^ theor, 6, 7 ; and this is the firft mention I have feco 
made of this law of the coefficients of the powers of a binomial, com- 
monly called Sir L Newton's binomial theorem, althc^ugh it is very 
evident that Sir Ifaac was not the firft inventor of it, the part of it 
properly belonging to him fccms to be only the extending of it to 
fradional indices, which was indeed an immediate eiFedl of the ge- 
neral method of denoting all roofs like powers with fradtional expo^ 
nents, the theorem being not at all altered. However it appears that 
our author Briggs was the firft who taught the rule for generating 
the .coefficients of the terms, fucceffively one from another, of any 
power of a binomial, independent of thofeof any other power. For 
having fhewn, in his Abacus Uayxf^ro^ (which he (6 calh on account 
l)f its frequent and excellent ufe, and of which a fmal) fpecimeq i^ 







Abacus nATXPHSTOZ. ^ 


1 


u 


G 


F 


£ 


D 


C 


B 


A 


MB) 


-® 


-t® 


+ ® 


^® 


-(i) 


+® 


® 


1 


I 


I 


1 


I 


k 


I 


t 


Q 


8 


7 


6 


5 


4 


3 






3$ 


28 


21 


'5 


10 6 








94 


5^ 


35 


30 


10 










126 


70 


u 


'5 








1 


126 


21 


6 






' 84 


^8 


7 








36 


8 
















9 



licre annexed,) that the numbers in the diagonal direftions, afccnding 
(tg^ rig;ht to left, atifi the coefficients of the powers pf i)inomials, the 



CONSTRUCTION OF 

indices being the figures in the iirfl: perpendicular coltinm A> which 
arealfo the coefficients of the 2d terms of each power (thofc of thp 
iirft terms being i, are here omitted ;) and fhatany one of thcfe dia-? 
gonal numbers is in proportion to the next higher in the diagonal^ a$ 
the vertical of the former ^s to (he marginal of,the latter, that is, as 
the uppermoft nprnt^er in the column of the former is to the firft qr 
right-hand number in the line of the latter; having (hewn thefe 
things, I fay, h^ thereby teaches the generation of the coefficients of 
any power, independently of all other powers, by the very fame law 
or rule which we now ufe in the binomial theorem. Thus, for the 
9th power I 9 being the coefficient of the 2d term, and i always that 
of the fifft, to find the 3d coefficient we have 2 : 8 : : 9 : 36 ; for the 
4lh term, 3 : 7 : : 36 : 84 ; for the 5th term, 4 : 6 : : 84 : 126 , aiid 
fo on for the reft. That is to fay, the coefficients of the terms in 
:|ny power si, are inverfely as the vertical numbers or firfi line i, 2^ 
3,4, .•••!», and dire<£lly as the afcending numbers »i,i»—i, |»— a» 
jfi— 3, • • • • I in the firft column A ; and ihat confeqqently thofe 
coefficients are found by the continual multiplication of thefe frac-? 

tions — »!LZl»!lZi> iZJ* — , which is the very theorem as it 

fiands at this day, and as applied by Newton to roots or fractional 
exponents, as it had before been ufed for integral powers. This 
theorem then being thus plainly taught by Briggs about the year 
1600, I am furprifed how a man of fuch general reading as Dr. Wal- 
lis v/as, Could poffibly be ignorant of it, as he plainly appears to be 
by the 85th chapter of his algebra, where he fully afcribes the inven-* 
tion to Newton, and adds that he himfelf had formerly fought after 
fuch a rule, but without fuccefs : Or how Mr. John Bernoulli, not 
half a century fince, could himfelf firft difpute the invention of this 
theorem with Newton, and then give the difcovery of it to M. Paf« 
cat, who was not born till long after it had been taught by Briggs. 
See Bernoulli's fForJtsy vol. 4 fa, 173. But I do not wonder tb;|i 
Briggs's remark was unknown to Newton, who owed almoft every 
thing tp genius, and very little to reading : and I have no doubt that 
he made the difcovery himfelf, without any light froni Briggs, anq 
that be thought it was new for all powers in general, as it was indeed 
for roots and quantities with fra£lional and irrational exponents. 

When the above table of the fums of figurate numbers is ufec| 
by our author in determining the coe^cients of the terms o fthe 
equation, whofe root is the chord of any fubmultiple^of an arc, as 
when the fedtion is exprcftld by any uneven number, he remarks that 
the powers of that c^ord or root will be the ;ft, 3d, 5th, 7th, &c, 
in the alternate uneven columns. A, C. £, G, &c, with their figns 
rf or — as marked to the powers, (continued till the higheft power 
be equal to the index of the feftion; and that the coefficients of 
thofe powers are the fums of two continuoys numbers in tlie fame 
column with the powers, beginning with i at the higheft power^ 
and gradually defcending one line obliquely to the right at each 



LOGARITHMS. 



n 



lower power : fo for a trife£lion, the numbers are i in C, and 1 + 9 
iPC 3 in A ; and therefore the terms are — i (^ + 3® : for a quinqui* 
fe£(ion, the numbers are i in E, j + 4= 5 in C, 2 4- 3 == 5 inA j 
fo that the terms are i@ — 5®+ 5 ® : for a feptife£(ion, the num** 
bers arc i in O, 1 + 6 = 7 m E, 4 + 10 = 14 in C, and 3 + 4 rs 
7 in A; and fo the terms are — 1® + y® — I4(D -h 7® : and fo 
on; the fum of all thefe terms being always equal to the chord of the 
whole or multiple arc. But^hen the. fe<^ion is denominated by an 
even number, the fquares of the chords enter the equation inftead of 
the Arft powers as before, and the dimenAons of all the powers are 
doubled, the coefficients being foiind as before, and therefore the 
'powers and numbers willbe thofe ip the 2d, 4th, 6th, &c, columns; 
^nd the uneven feAions may alfo be expreiTed the fame way : hence, 
for a bifeAion the terms will be — i@+4@; for a trife6lion 
J®— 6®+9(|>5 for thequadrifeaion— i(g)+8®— 2C®+i6@j 
for the quinquifefiion i@— io(^ +3S®'*5O0+25@ ; and fo on* 

Our author alfo fub- 
ioins another table, a 
fmall fpecimen of which 
is here annexed, in 
which the Arft column 
confifts of the uneven 
numbers, 1, 3, 5, &c, 
the reft being found by 
addition as before, and the alternate diagonal numbers themfelves^ 
are the coefficients. 

The method is quite different from 
that of Vieta, who gives another table 
for the like purpofe, a fmall part of 
which is here annnexed, which is 
formed by adding from the number a 
downwards obliquely towards the 
right ; and the coefficients of the terms 
ft^nd upon the horizontal line. 



+® 



+® 



7 



D 



I 



6 
20 



C 






i 

1+ 
30 



4 

9 

16 



A 
® 

t 



I 



3 
S 

7 

9 
It 



la 


I V 


ieta's JOi 


t 




3 


2d 




4 

s 


2 

5 


3d 




6 


9 


2 




7 


U 


I 


4th 


8 


20 


16 


2 


9 


27 


30 


9 


lO 


3S 


so 


*5 




Thefe angular feftlons were afterwards further difcuffed by 
Pughtred and Wallis. And the fame theorems of Vieta and Briggg 
have been fince given in a different form, by Meffrs, Herman, and the 
Bernoullis, in the Leipjic JSfSy and the Memoirs of the Royal Acade^ 
W of Sciences. Thefe theorems they cxpreffed by the alternate terms 
of the power of a binomial, whofe exponent is that of the multiple 
angle or fedion. And Mr, De Lagny, in the fame Memoirs, iirft 
Ihewed that the tangents and fecants of multiple angles are alfo ex- 

{^refled by the terms of a binomial, in the form of a fraflion, of which 
bmc of thofe terms form the numerator, and others the denominator. 
Tbus, if r exprefs the radius, s the fine, c the cofinc, / the tangent* 
^■^ • ^ * ' an4 



^9 CONSTRUCTION OF 

and /the fccant of the angle, J; then the fine, cofine, tangent, and 
fecant of n times the angle, are expreffcd thus, viz, 

* r"* ♦ 1.2.3 '•2-3'4»5 

-_- _ if ' • V . ^ «. «— 1 n*— 2 #* . n. «— r. n— 2. n — 3 » — ^4 4- 

TAnfrw^g^rs^ ^ . , L— - — — -^ ■ I 

r ~i.2 ■'"1.2.3,4 ^ * 

J' or r» -h /» 

1.2 *^l.2.3.4 * 

where it is evident that the feries in the fine oi nA confifts of the 

even jterms of the power of the binomial f+T^", and the fcries in the 
cofine of the uneven terms of the fame power ; alfo the feries in the 
numerator of the tangent confifts of the even terms of the power 
/•+/'"» ;ind the denominator, both of the tangent and fecant,. confifts 

of the uneven terms of the fame power r+/^». And if the diameter, 
chord, and chord of the fupplement, be fubftituted for the radius^ 
fme and cofine, in the expreffions for the multiple, fine and cofine, 
the refult will give the chord and chord of the fupplement of » times 
the arc or angle A* Thefe and various other exprei&ons for multi* 
pie and fubmultiplc arcs, with other improvements in trigonometry, 
have alfo been given by Euler and other eminent writers on the fuD- 
jea. 

The before mentioned M. De Lagny oiFered a proje6} for fubfti* 
tuting, inftead of the common logarithms, a binary arithmetic, 
which he called the natural logarithms^ and which he and M. Leib- 
nitz feeni to have both invented about the; fame time, independently 
of each other : but the projeft came to nothing. Mr. De Lagny alio 
publilhed, in feveral Memoirs of the Royal Academy, anew method 
of determining the angles of figures, which he called Goniomitry, It 
cpnfifts in meafuring with a pair of compafles the arc which fubtends 
the angle in queftion; however this arc is not meafured by applying 
its extent to any preconftruded fcale, but by examining what part it 
is of half the circumference of the fame circle, in this manner : 
^ from the propofed angular point as a center, with a fufficiently largQ 
radius, a femicircle being defcribed, a part of which is the arc inter- 
cepted by the fides of the propofed angle, the extent of this arc is 
taken with a fine pair of compafTes, and applied continually upon the 
arc of the femicircle, by which he finds bow often it is contained in 
the femicircle, with ufually a fmall arc remaining; in the fame nian- 
nerhe meafures how often this remaining arc is contained in the firft 
arc^ and what remains again is applied continually to the firft re-^ 



t O G A R I T H M S. 

I 

maikid^, and (o the 3d remainder to the 2d, the 4th to the jd, and fb 
on till there be no remainder, or elfe till it become infenfibly fmaU« 
fijr this procefs he obtains a feries of quotients, or fractional partb, 
one of another, which being properly reduced into one, give the ra- 
tio of the firfl arc to the femicircumference, or of the propofed angle, 
to two right angles or 180 degrees, and confequently that angle in 
degrees, minutes, &c, if required, and that commonly to a degree of 
accuracy far exceeding the calculation of the fame by means of any 
Cables of fines, tangents or fecants, notwichftanding; theapisarent pa- 
radox in this exprefiion at firft fight. Thus, if the ift arc be ^ 
times contained in the femicircle, the remainder once contained in 
the firft arc, the next five times in the fecond, and finally the fourth 
two times in the third : Here the quotients are4, i, 5, 2; confequent* 
]y the fourth or laft arc was i the 3d, therefore the 3d was-i^ or JL 

of the 2d, and the 2d was -^ or 11. of the ift, and the firft or arc 

II 

fought, was JL. or -li of the femicircle ; and confequently it con* 

477 
tains 37| degrees, or 37** 8^ 34''t- 

But to retarn from this long digrcffionj Mr. Briggs next treats of 
interpolation by differences, and chiefly of quinqui(e6lion, after the 
manner ufed in the 13th chapter of his conftruSion of logarithms 
before defcribed. He here proves that curious property of the fines 
and their feveral orders of differences, before mentioned, namely^, 
that, of equidifferent arcs, the fines, with the 2d, 4th, 6tn^ 
&c differences, are continued proportionals ; as alfo the cofines of 
the means between thofe arcs, and the ift, 3d, 5th, &c differences* 
And to this treatife on interpolation by differences, he adds a margi- 
nal note, complaining that this I3tfa cH^pterof hi8^ri7im^//Vtf Lega'^ 
ritbmica had been omitted by Vlacq in his edition of it; as if he were 
afraid of an intention to deprive him of the honour of the invention 
of interpolation by fucceifive differences. The note is this: Mcdu$ 
CQrre£lionis a me traditus eft Arithmetica Logarithmiea capiti 13, imdi*^ 
' thne Lonidnenji : Iftud autem caput una cumfequenti in editions Batava 
me inconfulto et inTcio omijjum fuit : nee in omnibus^ iditionis illius author 
vir alioqiii indujtrim et non indo^us rneam mentem videtur ajfiquutus : 
Ideoque ne quicquam defit cuiquam^ qui intfgrum canoriim (onfitere cupiat: 
queedam maxime necefjaria illinc hue transfer^nda cenjui. 

A large fpecimen of quinquifedition by differences is then given, 
and he mews how it is to be applied to the conftru£lion of the wh^e, 
canon of fines, both for looth and 1 000th part$ of degrees; namely^ 
forcentefms, divide the quadrant firft into 72 equ:|l parts, and find 
their fines by the primary methods; then thefe quinquifeded give 
360 parts, a fecond quinquifetftion gives 1800 parts, and a third 
gives 9000 parts, or centefms of degrees : but for millefms, divide the 
quadrant into' 144 equal parts; th^n one quinquife£Kon givj^ 720, 3 
fecond gives 3600, a third 18000, and a fourth gives 90COO parts, or 
jnillcfms, * • - . 

Ho 



%6 CONSTRlJCTldl^ 6f 

He next proceeds to the natural tangents and fecants^ whicis Hfi 
flireded to be raife'd in the fame manner, by interpolations from si 
few primary ones, conftrufted from the known proportions betweeri 
fines, tangents, and fecants ^ exceptkig that half the tangents and fe- 
cants are to be formed by addition andfubtraftion only^ by means of 
fome fuch theorems as thefe, namely, ift, the iecant of an arc is equal 
to the fum of the tangent of the fame arc, ftnd the tangent of half its 
complement, which will find every other fecant; ^d, double the tan- 
gent of an arc added to the tangent of half its complement, is equal to 
the tangent of the fum of that arc and the faid half complement^ by 
irhich rule half the tangents will be found; &c. 

In the two remaining chapters of this book ate treated the con* 
fini£Uon of the logarithmic fines, tangents, and fecants. This is 
preceded by fome remarks on the origin and invention of them. Our 
author here obferves that logarithms may be of various kinds; that 
others had followed the plan of Baron Napier the firft inventor^ 
amonr whom Benjamin Urfinus is efpecially commended, who ap« 
plied Napier's logarithms to every ten feconds of the quadrant ; but 
that he himfelf, encouraged by the noble inventor, deviled other !o* 
garithnts that were much eafier and more excellent*. He fays he 
put lo, with ciphers, for the logarithm of radius; 9 for the logar-a 
itbm fine of 5^ 44^ wbofe natural fine is oiie loth of tlfe radius ; g 
for that of 34^ whofe natural fine is one looth of the radius, Sect 
thercbY making i the logarithm of the ratio of 10 to if which is the 
charaocriftic of his fpecies of logarithms. 

To conftrud the logarithmic fines^ he directs firft to divide the 
quadrant into 72 equal parts as before, and to find the logarithms of 
their natural fines as in the 14th chapter of his ArithmetUa La- 
garubmi^ai after which this number will be increafed by quinqai&c- 
tton, firft to 360, then to 1800, and iaftly to 9000, or cenuXms of de« 
grees. But if millefins of degrees be required, divide the quadrant firit 
into 144 equal parts, and then by four quinquifedtions theie will bo 
caDfcfindedto the following parts, 720, 3600, 18000, and 90000, or mil -» 
lefim of degrees. He remarks however that the Ic^rthmic fines of 
only half the quadrant need be found in this manner, as the ofher 
hair may be found by mere addition, or fubtradion, bv means ot 
this theorem, as the fine of baif an arc is to half radius, lo is the fine 
of the whole arc to the cofine of the faid half arc. This theorem h^ 
illuftrates with examples^ and then adds a table of the logarthmic 
fines of the priiuary 72 parts of the quadrant^ from which the reft are 
to be made out by quinqui(e6lion. 

In the next chapter our author fhews the conftrudion of the natu« 
fal tangents -and fecants more fully than he had done before, de« 
anonftrating and illuftrating feveral curious theorems for the eaiV 
finding of them* He then concludes this chapter, and the book^ witn 



>••• 



^ His words are << Ego vero ipiius invemons primi cohortatiotie adjutus^ alios lo* 
farichmoc applicaados ceniiii, qui multo faoiliorem ufuitt tiabeat| prdtantiorem. Lo^ 
gahtfamut radii circularis vtl fiaus cotius, a mc ponitur 10 fcc." 

pointing 



LOGARITHMS, ti 

pointing out the very eafy conftruAion of the logarthmic tangeiiU 
and fecants by means of thefe three theorems : 

ifty As cofine : fine : : radius : tangent, 
2d, As tangent : radius : : radius : cotangent^ 
3d, As cofine : radius :: radius : fecant. 
So that in logarithms, the tangents are found by fubtracling the 
cofines from the fines, adding always 10 or the radius; the cotan- 
gents are found by fubtrading always the tangents from 20 or double 
the radius; and the fecants are found by fubtrading the cofines from 
20 the double radius. , 

The 2d book, by Gellibrand, contains the ufe of the canon ia 
plane and fpherical trigonometry. 

Beiides BHggs's methods of conflruf^ing logarithms, above de- 
fcribed, no others were given about that time. For as to the calcu« 
lations made by Vlacq, his numbers being carried to comparatively 
but few places of figures, they were performed by the eafiefl of 
Briggs's methods, and in the manner which this ingenious man had 
pointed out in his two volumes. Thus, the 70 chiliads of log^* 
rithms, from 20000 to 90000, computed by Vlacq, and publifhed in 
1628^ being extended only to 10 places, yield no more than two 
orders of mean differences, which arealfo the correft differences, ia 
quinquifedion, and therefore wilt be made out thus, namely, one^ 
fifth of them by the mere addition of the conflant logarithm of 5; 
and the other {bur-fifths of them by two eafy additions of very .fmall 
numbers, namely, of the i& and 2d differences, according to the di- 
reSions given in Briggs's Jrith. L^g. c. i}. p. 31. And as^to Vlacq's 
logarithmic fines and tangents to every 10 fecqnds, they were eafily 
computed thus; the fines for half the quadrant were found by 
taking the logarithms to the natural fines in Rheticus's canon; and 
then from thefe the logarithmic fines to the other half quadrant were 
found by mere addition and fubtradion ; and from thefe all the tan* 
gents by one iingle fubtradion. So that all thefe operations might 
eafily be performed by one perfon, as quickly as a printer could fet 
up the types ; and thus the computation and printing might both be 
carried on together. And hence it appears that there is no reafon 
for admiration at the expedition with which thefe tables were faid 
\o have been brought out. 

Of certain Curves related to Logarithms^ 

About this time the mathematicians of Europe began to confidef 
feme curves which have properties analogous to logarithms. Ed- 
mund Gunter, it has been faid, firfl gave the idea of a curve, whofe 
abfciffes are in arithmetical progreffion, while the correfponding or- 
dinates arc in geometrical progreffion, or whofe abfciffes are the log- 
arithms of their ordinates ; but I cannot find it noticed in any part of 
his writings.* The fame curve was afterwards confidered by others. 
Did named the (logarithmic or Logijiic curve by Huygens in his Dif^ 

M fertatif 



tt CONSTRUCTION OP 

fertatto'de taufa GravitattSy where h? enumerates all the principal 
properties of this curve, (hewing its analogy to logarithms. Many 
other learned men have alfo treated of its properties ; particularly 
LeSeur and Jacquier in their comment on Newton's Principia; Dr. 
John Kiell in the elegant little tra£l on logarithms fubjoined to his 
edition of Euclid's Elements ; and Francis Mafercs, Efq. Curfitor 
Bafon of the Exchequer, in his ingenious treat ife on Trigonometry; 
in which books the do£trine of logarithms is copioufly and learnedly 
treated, and their analogy to the logarithmic curve &c fully difplayed. 
*— It is indeed rather extraordinary that this curve was not fooner 
announced to the public ; fincc it refults immediately from baron 
Napier's nlanner of conceiving the generation of logarithms, by only 
fuppodng the lines which reprefent the natural numbers to be placed 
at right angles to that upon which the logarithms are taken. This 
curve greatly facilitates the conception of logarithms to the imagi* 
jiatioti, and aflbrds an almoft intuitive proof of the very important 
property of their fluxions, or very fmall increments, to wit, that the 
fluxion of the number is to the fluxion of the logarithm, as the number 
is to the fubtangent ; as alfo of this property, that, if thr^e nutnbers 
be taken very nearly equal, fo that their ratios to each other may 
diiFer but a little from a ratio of equality, as for example, the three 
numbers loooooco, loccoooi, 10000002, their differences will fe 
very nearly proportional to the logarithms of the ratios of thofe num* 
bers toeaoh other : all which follows from the logarithmic arcs being 
very little different from their chords, when they are taken very fmalL 
And theconftant fubtangent of this curve is what was afterwards by 
Cotes called the Modulus of the fyftem of logarithms : and ilnce, by 
the former of the two properties abovementioned, this fubtangent is 
a 4th proportional to the fluxion of the number, the fluxion of the 
logarithm, and the number, this property afforded occafion to Mr. 
Baron Maferes to give the following definition of the modulus^ 
which is the fame in effe£t as Cotes's, but more clearly expreffed, 
ramely, that it is the limit of the magnitude of a 4th proportional 
to thefe thr^e quantities, to wit, the difference of any two natural 
numbers that are very nearly equal to each other, either of the faid 
numbers, and the logarithm or meafure of the ratio they have to 
each other. Or we may define the modulus to be the natural num- 
Ijer at that part of the fyilem of logarithms, where the fluxion of the 
number is equal tp the fluxion of the logarithm, or where the numbers 
and logarithms have equal diifefences. And hence it follows, that 
the logarithms of equal numbers or of equal ratios, in different fyi^ems, 
are to oneanother as the tpoduli pf thofe fyftems. Moreover, the ra- 
tio whofe meafure or logarithm is equal to the modulus, and thence ' 
by Cotes called the ratio modu/ansy is by calculation found to be the 
ratio of 2718281828459 &cto I, or of I to '367879441 171 &c; the 
calculation of which number may be feen at f'ull length in Mr. baron 
Maferes's treatife on the Principles pf l^iife-annuities^ pa. 274 and 

^ Tho 



LOGARITHM S« 8j 

't'he hyperbolic cun'c alfo afforded another fburce for developing 
and illuftrating the properties and conftru6bion of logarithms. For 
the hyperbolic areas lying between the curve and one afymptote, 
when they are bounded by ordinates parallel to the other afymptote^ 
are analogous to the logarithms of their abfcifles or parts of the 
afymptote. And fb alfo are the hyperbolic fedors ; any feftor 
bounded by an arc of the hyperbola and two radii^ being equal to 
the quadrilateral fpace bounded by the fanne arc, the two ordiilates 
to either afymptote from the extremities of the arc, and the part of 
the • afymptote intercepted between them. And although Napier's 
logarithms are commonly faid to be the fame as hyperbolic logar-> 
ithms^ it is not to be underftood that hyperbolas exhibit Napier's 
logarithms only, but indeed all other poflible fyftems of logarithms 
whatever. For, like as the right-angled hyperbola, the ftdeof whofe 
fquare infcribed atthevertex is i, gives us Napier's logarithms; foany 
other fyftem of logarithms is expreiTed by the hyperbola whofe afymp- 
totes form a certain oblique angle, theiide of the rhombus infcribed 
at the vertex of the hyperbola in this cafe alfo being ftill i, the fame 
as the iide of the fquare in. the right-angled hyperbola^ But the 
areas of the fquare and rhombus, and confequentiy the logarithms of 
'any one and the fame number or ratio, will differ according to the 
fine of the angle of the afymptptes. And the area of the fquare or 
rhombus, or any infcribed parallelogram, is alfo the fame tl^ng as 
what was by Cotes called the modulus of the fyftem of logarithms ; 
which modulus will therefore be expreiTed by the numerical meafure 
of the fine of the angle formed by the afymptotes, to the radius i ; 
as that is the fame with the number expreffing the area of the faid 
fquare or rhombus, the fide being i : which is another definition of 
the modulus to be added to thofe we before remarked above in treat- 
ing of the logarithmic curve. And the evident reafon of this is^ that 
in the beginning of the generation of thefe areas from the vertex of 
the hyperbola, the nafcent increment of the abfcifTe drawn into the 
altitude i, is to the increnAent of the area, as radius is to thefine of the 
angle of the ordinate and abfciffe^ or of the afymptotes ; and at the 
beginning of the logarithms, the nafcent increment of the natural 
numbers is to the increment of the logarithms, as i is to the modu- 
lus of the fyftem» Hence tie eafily difcover that the angle formed by 
the afymptotes of the hyperbola exhibiting Briggs^s fyftem of logar- 
ithms, will be 25 deg. 44 mtn. 25I fee. this being the angle whofe 
fine is 0*4342944819 &c, the modulus of this fyftem. 

Or indeed any one hyperbpla, as has been remarked by Mr, baron 
Maferes, will exprefs all poflible fyftems of logarithms whatever^ 
namely, if the fquare or rhombus infcribed at the vertex, or, which 
is the fame thing, any parallelogram infcribed between the afymp* 
totes and the curve at any other point, be expounded by the modulus 
of the fyftem ; or, which is the fame, by expounding the area, in« 
tercepted between two ordinates which are to each other in the ratio 
cf 10 to i> by the logarithm of that ratio in the propofed fyftem. 

Ma At 



84 CONSTRUCTION OF 

As to the firft remarks on the analogy between logarithms and 
the hyperbolic fpaces j it having been ihewn by Gregory St. Vincent, 
in his ^uadratura Circuit isf Se^ionum Coni^ publifhed at Antwerp in 
1647, that if one afymptote be divided into parts in geometrical 
progref&on, apd from the points of divifion ordinates be ((rawn pa- 
.rallel to the other afymptote, they will divide the fpace between the 
afymptote and curve into eaual portions ; from hence it was (hewn 
by Meriennus, that, by taking the continual fums of thofe parts, 
t^iere would be obtained areas in arithmetical progreffion» adapted to 
abfbiiles in geometrical progreffion, and which therefore were ana- 
logous to a fyftein of logarithms. And the fame analogy was re- 
marked and illuftrated foon after by Huygens and many others, who 
ihew how to fquare the h;^ perboHc fpaces by means of the logarithma* 

Of^Gr<g9ry*s Computathn of Logarithmim 

On the other band, Mr. James Gregory, in his Vira Circuit and 
JHyperhlig ^uadratura^ firft printed at Patavi, or Padua, in the year 
1667, having approximated to the hyperbolic afvmptotic fpaces by 
means of a feries of infcribed and circumfcribed polygons, from thence 
ihews how to compute the logarithms, which are analogous to thofe 
^reas : and thus the quadrature of the hyperbolic fpaces became the 
fame thing as the computation of the logarithms. He here alfo lays 
down various methods to abridge the computation, with the affift- 
ance of fome properties of numbers themfelves, by. which we are 
enabled to compofe the logarithms of alt prime numbers under lOOO, 
each by one multiplication, two divifions, and the extraSjon of the 
fquare root. And the fame fubje£t is farther purfued in his Exerci- 
tatifffies Geomctriae^ to be 4efcribed hereafter. 

There are alfo innumerable other geometrical figures having pro- 
perties analogous to logarithms \ fuch as the equiangular fpiral, the 
figures of the tangents and fccants, &c ; which it is not to our pur- 
pofe todiftinguilh more particularly. 

Of \MercatGr*5 L^garithmotecbnia, 

In 1668, Nicholas Mercator publifhed his Logarithmotechnia^ fi^e 
methdus cenjiruendi Logariihmos nova^ accurata^ {9* facilis ; in which 
he delivers a new and ingenious method for computing the logarithms 
upon principles purely arithmetical ; which being curious and very 



•«■ 



* James Gregory was bom at Aberdeen in Scotland 1639, where be was educated. 
He was profeflbr of Mathematics in the college of St. Andrews ; and died of a fever in 
December 167-5, being only 36 years of age. 

f Kicholas Mercator, a learned mathematictani and an ingeaioos member of Che 
Royal Society, was a native of Holftein in Gci-many, but fpent moft of his time in ^og-^ 
landi where he died in the year 1690, at about 50 yejtrs of age. He was the anchor of 
XMifij Other works in GeomeCryi Geography^ Aftvonomyi Aftrdogy, &c. 

accurately 



LOGARITHMS. 9$ 

aiecurately per(otmei, I (hall here give a rather AiII and particular 
account of that little trad» as well as of the fmall fpecimen of the 
quadrature of curves by Infinite feries fubjoined to it ; and more ef* 
pecially as this work gave occafion to the public commuaicatioH of 
ibme of Sir Ifaac Newton's earlieft pieces, to evince that he had not 
borrowed them ffooEi this publication. So it appears that thefe two 
ingenious men bad, independent of each other, in fome inftances 
fallen upon the fanoe things. 

Our author begins this work with remarking that the word Lofar* 
ithm is compofed of the words ratio and numbir^ being as much as to 
fay t\i!i number of ratios ; which he obferves is quite agreeable to the 
nature of them, for that a logarithm is nothing elfe but the nombec 
of ratiuncul^ contained in the ratio which any number bears to unity. 
He then makes a very learned and critical diiTertation on the nature 
of ratios, their magnitude and meafure, conveying a clearer idea of 
the nature of logarithms than had been given by either Nie^er or 
Briggs, or any other writer except the famous Kepler, in his woiic be* 
fore defcribed, although thoie other writers feem indeed to have had 
in their own mfnds the fame ideas on the fubjeA as Kepler and Mer« 
cator, but without having exprefled tjbem fo clearly. Our author 
indeed pretty clofely follows ICepIer in his nKides of thinking and 
expreffion, and after him in plain and exprefs terms calls logarithms 
the meafures of ratios ; and, in order to the right underftanding that 
definition of them, he explains what he means by the magnitude of a 
ratio* This he does pretty fully, but* not too fully, confiderihg the 
nicety and fubtlety of the fubje£l of ratios, their magnitude, with 
their addition to, and fubtradion from» each other, which have been 
mifconceived by very learned mathematicians, who have thence been 
led into confiderable miftakes. Witnefs the overfight of Gregory 
St. Vincent, which Huygens animadverted upon in the E|rra#i( C/» 
thmitri^ Gngorii a San^o Vincfntioy and which arofe from not under^ 
ftanding, or not. adverting to, the nature of ratios, and their propor« 
tions one to another* And many other fimilar miftakes mi^t here be 
adduced of other eminent writers. From ajl which we muft com- 
mend the propriety of our author's attention, in fo judicioufly^ difcri« 
minating between the nwgnitude of a ratio, as of «/to ^, suid the 

fraAion • , or quotient arifing from the divifion of one term of the 

ratio by the other ; which latter method of confideration is always 
attended with danger of errors and confufion on the fufcjefl; though 
in the 5th definition of the 6th book of Euclid this quotient is ac-» 
counted the^quantity of- the -ration but-thMdefinttien isprobabhr not 
genuine, and therefore very properly omitted by profeflor Simion in 
his edition of the Elements. And in thofe ideas on the fubjed of 
logarithms, Kepler and Metcator have been followed by Halley^ 
Cotes, and moft other eminent writers fince that time. 

Purely from the above idea of logarithms, namely as being the 
meafures of ratios, and as expreffing the number of ratiuncula con- 
tained in any ratio, or into which it may be divided^ the number of 

the 



66 C0NSTRiJctI0N6F 

the like equal ratiumula cont2t.\ned in fome one ratio, as of i6 to ti 
bein^ fuppofed given, our author (hews how the logarithm err mea- 
fure of any other ratio may be found. But this hourever only by* 
the-^bye, as not being the principal method he intends to teach, a« 
bis laft and beft, and which we arrive not at till near the end of the 
book, as we (hall fee b^low. Having ihewn then, that thefe logar- 
ithms, or numbers of fmall ratios^ or meafures of ratios, may be all 
properly reprefented by numbers, and that of i, or the ratio of equa- 
lity, the logarithm or meafure being always o, the logarithm of 
lO, or the meafure of the ratio of lo to i, is moft conveniently repre- 
fented by I with any number of ciphers ; he then proceeds to fhevtr 
bo W the meafures of all other ratios may be found from this laft fup« 
poiition* And he explains the principles by the two following ex- 
amples. 

Firft, to find the logarithm of ioo'5*, or to find how many raii^ 
ji?t€ulazvt contained in the ratio of 100*5 ^^ ^y ^^^ number of ratiuft'* 
OiUt in the decuple ratio, or ratio of 10 to i, being 1,0000000. 

The given ratio 100*5 to 1, he firft divides into its parts, namely 
iCO«5 to iO0» 100 to 10, and 10 to 1 ; the laft two of which being 
decuples, it follows that the chara6teriftk: will be 2, and it only re* 
mains to find how many parts of the next decuple belong to the firft 
Mtio of XOO'5 to 100. Now if each term of this ratio be multiplied 
by itfelf, theproduds will be in the duplicate ratio of the firft terms, 
or this laft ratio will contain a double number of parts ; and if thefe 
be multiplied by the firft terms again, the ratio of the laft produds 
will contain three times the number of parts ; and fo on, the num- 
ber of times of the firft parts contained in the ratio of any like poiVers 
of the firft terms, being always denoted by the exponent of the power. 
If therefore the firft terms, ioO'5 and 100, be continually multiplied 
till the fame powers of thenl have to each other a ratio whofe meafure 
18 known, as fuppofe the decuple ratio 10 to i, whofe meafure is 
1,0000000; then the exponent of rhat power (hews what multiple 
tbis meafure 1,0000000, of the decuple ratio, is of the required mea- 
fure of the firft ratio 100-5 ^^ '^O ; and confequently dividing 
1,0000000 by that exponent, the quotient is ihe meafure of the ratio 
lOO'S to 100 fought. The operation for finding this, he fets down 
as here follows; where the feveral multiplications are all performed 
in the contrafted way by inverting the figures of the multiplier, and 
xetaining only the firft number of decimals m each product. 

powei: 



^ Mercator diftln{;uilhes his decimals from iategers thus ioo[5, or thus 10015. 



LOGARITHMS. 



87 



100*5000 
5001 

1005000 

I 01 002 5 
5200101 

1010025 

JOIOQ 
20 

5. 

I 0201 50 

O5IO2OZ 

■■•^^^^■■^^^^ 

1 020 1 50 
IQZ 

£1 

1040706 
607040 r 

1083068 
8603801 

53037'J 
1376011 
1106731 

1893406 
6043981 

3584985 

5894853 

1^852116 

Since 



power 

- I 



2 
z 



'4 
4 



. 8 
. 8 

. 16 
. 16 

• 3^ 

• 32 

- 64 

- 64 
128 
128 

256 
^56 

512 



This power being 
greater than the decuple 
of the like power of 100, 
which muft always be i 
with ciphers, refume 
therefore the 256th 
power, and multiply it, 
not by itfelf, but by the 
next before it, viz. by 
the i28tb, thus 



3584985 
6043981 

6787831 
1106731 

9340730 
5303711 



10956299 



256 
128 

384 
64 

448 
32 

>8o 



This power again ex- 
ceeding the fame power 
of lOQ more than 10 
times, I therefore draw 
the fame 448th, not in- 
to the 3 2d, but the next 
preceding, thus 



9340130 
8603801 

101 1 5994 



- 448 

- 16 

- 464 



Thii being again too 
much, inftead of the 
i6tb, draw it into the 
8th, or next preceding, 
thus 



9340130 
6070401 

9720329 
O5IO2OI 

99I6I93 
52OOIOI 



• 448 

- . 8 

- 4S^ 

4 

• 460 

^ 462 



10015603 « «i 

Which power agaia 
exceeds the limit ; there- 
fore draw the 460th into 
the I ft thus 

9916193 - . 460 
5001 - . I 

9965774 - . 461 

Since therefore the 
46 ad power of 100*5 '* 
greater, and the 461ft 
power is lefs, than the 
decuple of the fame 
power of 100; I find 
that the ratio of 100*5 
to 100 is contained, in 
the decuple more than 

461 times, but lefs thaa 

462 times. Again, 



99i6i93')and the difFerences 



incef 460-) r 99i6i93|^nd the difFerei 

the 1 +^' f IS 1 9965774 K9581 f nearly 
(. 462 J (_ 100 1 5603 J 49829 1 equal ; 

therefore the proportional part which the exaft power, or lOOOOOOO, 
exceeds the next lefs 9965774, will be eafily and accurately found 
hy the Golden Rule, thus : 

The juft power - • - locooooo 
and the next lefs - -• 9965774 

the difference - - 34226 j then 

As 49829 the dif. between the next lefs and greater, 
: To 34226 the d^f. between the next lefs and juft, 
: : So IS loooo: to 68689 the decimal parts; and therefore the ra* 
tioof 100*5 ^^ ^0^9 ^^ 461*6868 times contained in the decuple or 
ratio of 10 to I. Dividing now 1,0000000, the meafurq of the de- 
cuple ratio, by 461*6868, the quotient 00216597 is the meafure of 
the ratio of 100*5 to ico^ which being added to 2 th,e meafure of 



<» CONSTRUCTION OF 

100 to I, the fum 2,00216597 Is the meafure of the ratio of 100*5 'to 
I, that is the log. of 100*5 *^ 2,00216597. 

In the fame manner he next invtftigatcs the log, of 99*5, and finds 
it to be 1,99782307. 

A few oblervations are then added, calculated to generalize the 
confideration of ratios, their magnitude and afFe^ions. It is here 
remarked that he confiders the magnitude of the ratio between two 
quantities as the fame, whether the antecedent be the greater or the 
lefs of the two terms : fo the magnitude of the ratio of 8 to 5, is 
the fame as of 5 to 8 ; that is by the magnitude of the ratio of either 
to the other, is meant the number of ratiuncula between them, 
which will evidently be the fame whether the greater or lefs term 
be the antecedent. And he farther remarks that of different ratios, 
when we divide the greater term of each ratio by the lefs, that ritio 
is of the greater mafs or magnitude which produces the greater quo- 
tient, tt via verfa ; although thofe quotients are not proportional 
to the maifes or magnitudes of the ratios. But when he confiders 
the ratio of a greater term to a Ie6, or of a lefs to a greater, that is 
to fay, the ratio of greater or lefs inequality, as abftra<£hd from the 
magnitude of the ratio, he diftinguifhes it by the word offeStiw^ as 
much as to fay greater or lefs affedion, fomething in the manner of 
pofitive and negative quantities, or fuch as are affefted with the 

figns -|- and — The remainder of this work he delivers in 

feveral propofitions, as follows. 

Prop. I. In fubtrafting from each other two quantities of the 
fame afiedion, to wit, both pofitive, or both negative ; if the remain- 
der be of the fame affefiion with the two given, then is the quantity 
fubtraded the lefs of the two, or exprefled by the lefs number \ but if 
the contrary, it is the greater. 

Trap. 2. In any continued ratios, as — ^> f[lt£.> f[Z5f-> &c,(by 

tf+^ «+2^ tf+3^ 
which is meant the ratios oi a to tf+^, a-k-h to tf+2^, a'Y^h to 
tf+3^» &c,) of equidifFerent erms, the antecedent of each ratio 
being equal to the confequent of the next preceding one, and pro- 
ceedins from lefs« terms to greater ; the meafure of each ratio will be 
exprefled by a greater quantity than that of the next following \ and 
the fame through all their orders of differences, namely, the ift, 2d^ 
3d, &c, differences) but the contrary when the terms of the ratios 
decreafe from greater to lefs. 

Prop. 3. In any continued ratios of equidifferent terms, if the 
ift or leaft beu, the difference between the ift and 2d *, and f,^, e^ 
&c, the refpedive firft term of their 2d, 3d, 4th, &c, differences ; 
then (hall the fevera} quantities tbemfelves be as' in the annexed 

fphemts 



LO OAR I TH MS. 



89 



I ft term - 
2d - - 

3d 

4th - - 
Sth - - 



a+ h 

a + 2b -^ e 

« +3* + 3c + d 

a-ir 4^ + 0C + 4^/+ * 




fcheme; where each term is 
compofed of the firft term to- 
gether with as many of the dif- 
ferences as it is diftant fromf 
the firft term, and to thofe dif- 
ferences joining, for coeffici- 
ents, the numbers in the Hop- 
ing or oblique lines contained 
In the annexed table of figu- 
rate numbers, in the fame 
manner, he obferves, as the 
fame figurate numbers com- 
pleat the powers raifed from a 
binomial root, as had long be- 
fore been taught by others. 
He alfo remarks that this rule 
not only gives any one term, 
but alfo the fum of any num- 
ber of fucceffivc terms from 
the beginning, making the 
2d coefficient the firft, the 3d 
the 2d, and fo on ; thus, the 
fum of the firft 5 terms is $a + 10b + 10c + 5^ + ^» 

In the 4,thprop. it is fhewn 
that if the terms decreaie, 
proceeding from the greater 
to the lefs, the fame theo- 
rems hold good, by only 
changing the fign of every 
other term, as in the margin. 

Prop. 6 and 7 treat of the approximate multiplication and divi- 
fion of ratios, or, which is the fame thing, the finding nearly any 
powers or any roots of a given fraction, in an eafy manner. The 
theorem for ralfing any power, when reduced to a Ampler form, 

is this, the m power of ±, or 'L; is — !Si^ nearly, where / is=: 

a + if and d:=: a ^ *, the fum and difference of the two numbers, 

and the upper or under figns take place according as j- is a proper 
or an improper fraction, that it according as a is lefs or greater 
than b» And the theorem for cxtradling the mth root of yis *7 —or 

T|^z:^!ix£ nearly; which latter rule is alfo the fame as the for- 
mer, as will be evident by fubftituting - inftead of m in the firft 



liiterm - - « 

2d - . - a^^ b 

3d • — - tf — 2^ + V 

4th - - - tf— 3^+3^— ^ 

5th ^ - - tf — 4A + 0f— 4rf+# 
&c. &c. 



m 



theorem. Sp that univerfally-7|~«=^ii^ nearly. Thefe theo. 



N 



rcms 



^ CONSTRUCTION OF 

rems however arc nearly true only in feme certain cafes, namely 
when -1 and ^ do not differ greatly from unity. And in the yth 

prop^ the author (hews how to find nearly the error of the theo- 
rems. 

In the 8th ^ro^. it is (hewn that the meafures of ratios of equidif- 
fercnt terms, are nearly reciprocally as the arithmetical means be- 
tween the terms of each ratio. So of the ratios i|, A|, 15, the mean 
between the terms of the firft ratio is 17, of the ad 34, of the 3d 51, 
and the meafures of the ratios are nearly as i^, J^, JL. 

From this property he proceeds, in the 9th prop, to find the mea- 
fure of any ratio lefs than J!21f, which has an equal difference (i) 
of terms. In the two examples, mentioned near the beginning, our 

author found the logarithm or meafure of the ratio, of 22:£ tobe 

^ 100' 

21769^) and that of j|^ to be 21659/5 ; therefore the fum 43429 

is the logarithm o(-~jr» ^^ "?^ ^ 1:^$'* ^^ ^^^ logarithm of -^5^ 
is nearer 43430, as found by other more accurate computations. — 
Now to find the logarithm of •^, having the fame difference of terms 
(i) with the former; it will be, by prop. 8, as 100.5 (the mean 
between 101 and 100) : 100 (the mean between 9Q'5 and ico'5) 
• ' 43430 : 43213 the logarithm of -^, or the difference between 
the logarithms of loo and loi. But the log. of 100 is 2 ; therefore 
the logarithm of loi is 2,0045213.— —Again, to find the loga- 
rithm of 102, we muff firft find the logarithm of -^^ the mean be- 
tween its terms being 101 '5, therefore as ioi'5 : 100 : : 43430 : 
42788 the logarithm of -J^, or the difference of the logarithms of 
10 1 and lOZ* But the logarithm of 10 1 was found above to be 
^,0043213 ; therefore the logarithm of 102 is 2,0086001. — So that 
dividing continually 868596 ( the double of 434298 the logarithm of 
-j—^or i||-) by each number of the ftries 201, 203, 205, 207, &c, 
then add 2 to the ift quotient, to the fum add the 2d quotient, and fo 
on, adding always the next quotient to the laft fum, the feveral 
fums will be the refpeflive logarithms of the numbers in this fertes 
101, 102, 103, 104, &c. 

The next, or prop. 10, (hews that, of two pair of continued ratios 
whofe terms have equal differences, the difference of the meafures of 
the firft two ratios, is to the difference of the meafures of the other 
two, as the fquare of the common term in the two latter, is to that 

in the former, nearly. Thus, in the four ratios -iL > —-,9 

^+3^ . «+4^ - ^ ^aa+2aB 

T^h ^h ^5 ^ft^ meafure of '==^^ (the difference of the firft 

two, or the quotient of the two fraflions ) ; the meafure of 

--rTj^z • = ^+4^ • VR^S nearly. 

In 



LOGARITHMS. 9t 

In prop. II the author Ihews that fimilar properties take place 
among two fets of ratios confining each of 3 or 4 &c continued num- ^ 
bers. 

Prop. 12 ihews that, of the powers of numbers in arithmetical 
proereffion, the orders of differences which become equal, are the 
2d differences in the fquares, the 3d differences in the cubes, the 4th 
differences in the 4th powers, &c. And from hence it is (hewn how 
to conftrud all thoie powers by the continual addition of their dif- 
ferences. As had been Long before more fullv e3q>lained by Briggs. 

In the next, or 13th prop, our author explains his compendiou« 
method of raifing the tables of logarithms, (hewing how to conftrud 
the logarithms by addition only, from the properties contained in the 
8th, 9th, and 12th propofitions. For this purpofe he makes ufe of 

the quantity -^, which by divifion he refolves into this infinite fe- 

Ties ^ +|j +ji + f^ &c (in tnfin.) Putting then « = 100 the arith- 
metical mean between the terms of the ratio -—^j *=riooooo> 
and c fucceffively equal to 0*5, i*<, 2*5, &c, that fo A— r may be re- 
fpe^iively equal togQggg'S, 99990*5^ 99997'S» &c» the correfpondihg 
means between the terms of .the ratios •.^^^, ||^, |||fl, &c, it 

IS evident that -^ will be the quotient of the 2d term divided by the 

iff in the proportions mentioned in the 8^ and 9th propofitions; 
and when each of theie quotients are found,, it remains then only to 
multiply them by the conftant 3d term 43429, or rather 43429*89 
of the proportion, to produce the logarithms of the ratios 
.99929. 99991, 99997 ^ jj,j 1222? . then adding thefe conti- 

«cxx>oo 9 9999 99998 loooi ° 

nally to 4 the logarithm of loooo the leaft number, or fubtrading 
them from 5 the logarithm of the higheft term looooo, there will 
refult the logarithms of all the abfolute numbers from lOOOO to 
looocx). Now when cis zz 0,5, then .^^^^ »}^^ 

"T^'OOl, ■7?=*000000005, —tl '000000000000025, -7J*="OOOOOOOOOOOOOOOOOOI25, 

It ^ 



Jcc; thtrefore -— - ^---^--+._ jj^ is = •001000005000025000125, 
In like manner, if c^i.j, then 7— will be =:-ooiooooi5ooo£25oo3375 1 

and iff r:2'5, then — — wUlbe ::^ •00x000025000625015625 5 
&c. But inftead of coAftrufting all the values of ..1- j^^ ^^e yfy^j 
way of raifing the powers, he dircds them to be fjund by addi- 

N » tion 



M 



CONSTRUCTION OF 



t!on only, as in the laft propofiion. Hftving thus 
found all the values of .^i-, the author then fhews 



b-c 



3 

4 

5 
6 

7 
8 



434*9 
86858 
130287 

'737^6 

217145 
260574 

3040O3 

347432 
390861 



that they may be drawn into the conftant logarithm 
43429 by addition only, by the help of the annexed 
table of the iirft 9 produds of it. 

The author then diftinguifhes which of the 
logarithms it may be proper to And in this wav, 

and which from their component .parts. 6f 

thefc the logarithms of all even numbers need not be thus 
computed, being compofcd from the number 2 j which cuts off one 
half of the numbers : neither are thofe numbers to be computed 
which end in 5, becaufe 5 is one of their factors 5 thefe laft are ' 
of the numbers ; and the two together i+^^ make | of the whol" 
and of the other 4, the \ of them, or ^ of the whole, are compofed 
of 3 ; and hence |+tt> or i\ of the numbers, are made up of fuch 
as are compofed of 2, 3, and 5, As to the other numbers which may 
be compofed of 7, of 11, &c j he recommends to find /A^/r logarithms 
in the general way, the fame as if th^y were incompofites, as it is not 
worth while to feparate them in fo eafy a mode of calculation. So 
that of the 90 chiliads of numbers from loooo to 1 00000, only 24. 
chiliads are to be computed. Neither indeed are all of thefe to he 
calculated from the foregoing feries for — f-, but only a few of 

them in that way, and the reft by the proportion in the Wi.propo- 
fition. Thus having computed the logarithms of 10003 and 10013, 
omitting 10023 as being divifible by 3, eftimate the logarithms of 
10033 and 10043, which are the 30th numbers from IC003 andiooi3; 
and again omitting icfo5j, k multiple of 3, find the logarithm* 
of X0063 and 10073. Then by prop. 8, 

As 10046, the arithmetical mean between 10033 ***d 10065, 

to 10018, the arithmetical mean berween 10:03 *nd IC033, 

fo 1 3006, the difFerence between the logarithms of 10003 and 10033, 

to 12967, the dificrence between the logarithms of 10033 and 10065 5 



10048 1 
Thatis, ift - -- As^ 10078?: 10018 



AgaiD, As 



And 3dly 



>f 




10028 



13006 



12992 



02967 

A &c. 

U &c. 



10038 : : 12979 



f 



2940 
&c. 



[,10068^ 
I 0098 > 
^ &c 3 
And with this our author concludes his compendium for con- 
ftruding the tables of logarithms. 

He afterwards fhews fome applications and relations of the doc- 
trine of logarithms to geometrical figures : in order to which, in 

prop. 



N 




LOGARITHMS. ^f 

prep* 14 be proves ailgebraically that, in the right- 
angled hyperbola, if from the vertex and from any 
other point there be drawn BI, FH perpendicular to 
the afymptote AH, or parallel to the other afymp« 
tote 5 then will AH : AI : : BI : FH. And 
In^0>^.i5,if AIz:Bl = i, and Hl^a; then will 

FH=: 7Tj'=i— <J+a*— tf'+rf*— a* &c in infinitum^ 

by a continual algebraic divifioh, the procefs of which he defcribes 
^p by ftep as a thing that was new or uncommon. But that 
method of divifion had been taught before by Dr. Willis in his Opui 
Jlritbmiticum* 

Prop, 16 is this : Any given number being fuppoled to be divided 
into innumerable fmall equal parts, it is required to affign the Aim 
of any powers of the continual fums of thofe innumerable parts* 
For which purpofe he lays down this rule; if the next higher 
power of the given number, above that power whofe fum is fou^t, 
be divided by its exponent, the quotient will be the fum of the 
powers fought. That is, if ff be the given number, and a on^ of 
its innumerable equal parts, then will 

if +I3Y'+P'+"5)»&C .... JV^ber: 11 : which theorem he de- 

monftrates by a method of induaion. And this, it is evident, i§ 
the finding the fum of any powers of an infinite number of arith- 
meticals, of which the greatcft term is a given quantity, and the 
leaft indefinitely fmall. It is alfo remarkable that the above exprcf- 
fion is fimilar to the rule for finding the fluent to the given fluxion 
of a power, as afterwards taught by Sir I. Newton. 

Our author then applies this rule in prop. 17, to the quadrature 
of the hyperbola. Thus, putting Al = i, conceive the afymptote to 
be divided from I into innumerable equal parts, namely Ip=:pq=rnr 
=« ; then, by the 14th and 15th, 

qt =i-2«+ 4<»*~ %ai &c^ i"* ^^ area BIruis=the fum ps+qt 

3— 6fl-}-i4a»— 36<73 &c, that is, equal to the number of terms 
contained in the line Ir, minus the fum of thofe terms, plus the 
fum of the fquares of the fame, minus the fum of their cubes, plus 
the fum of the 4th powers, &c. Putting now lA = i, as before, and 
Ip=o«i the number of terms, to find the areaBIps ; by prop. 16 the 

fum of the terms will be i^' = -005, the fum of their fquaress 
•000333333, the fum of their cubes 000025, «he fum of the 4th po- 
wers = -000002, the fum of the 5th powers =-000000166, the fum 
of the 6th powers =-000000014, &<:• Therefore the area BIps is 
- I —-005 + -000333333 — .(J00025 + -000002-- •000000166 + 
•000000014&C. = -100335347 —005025166 = -0953*0181 &c. 

Again, puttmg la =-21 the number of terms, he finds iji like 
manner the axea BIqt= -21-, -02205+ -003087 —000486202 + 

*oooo8i68a 



94 CONSTRUCTION OF 

•000081682 — *ooooi4294 + •000002572 — -000000472 + 
^00000088 &c =: *2i^i7i34S *- '022550084 = 7190620361 &c# 

He then adds, hence it appears that, as the ratio of AI to Ap, or 

I to I 'I, is half or fubduplicate of the ratio of AI to Aq^ or i to 

1^21, fo the area BIps is here found to be half of the area BIqt* 

Thefe areas he computes to 44 places of figures, and finds them 

'flill in the ratio of 2 to i. 

The foregoing dodrine amounts to this, that if the redangle BI 
y, Ir, which in this cafe is exprefled by Ir only, be put =r ^, AL be^ 
ing IT I as before) then the area BIru, or the hyperbolic logarithm 
4>f I + i/, or of the* ratio of i to i + i^^ will be equal to the infinite 
feries A — iA^ + 1-<#»— j-<#* + {A^ &c ; and which therefore may be 
confidered as Mercator's quadrature of the hyperbola, or his gene- 
ral expreffion of an hyperbolic logarithm in an infinite feries. And 
this method was farther improved by Dr. Wallis in the PhiloC 
Tninf. for the year 1668. 

In prop. 18 our author compares the hyperbolic areola with the r^i- 
ituncuUe of equidifFerent numbers, and obferves that 
the areola BIps is the meafure of the ratiuncula of AI to Ap* 
the areola spqt is the meafure of the ratiuncula of Al to Aq, 
the areola tqru is the meafure of the ratiuncula of Aq to Ar, &c. 

Finally, in the 19th prop, he fhews how thefums of logarithms 
may be taken after the manner of the fums of the areola* And from 
hence infers as a corollary, how the continual produd of any given 
numbers in arithmetical progreffion may be obtained : for the 
fum of the logarithms is the logarithm of the continual produd. 
He then remarks that from the premifes it appears in what manner 
Merfennus's problem may be refolved, if not geometrically, atleaft 
in figures to any number of places* And thus clofes this ingenious 
tra& 

In the Philof. Trzuf. (or 1668 are alfo given fome farther illuftra- 
tions of this work by the author himfelf. And in various places alfo 
in a fimilar manner are logarithms and hyperbolic areas treated of by 
Lord Brouncker, Dr. Wallis^ Sir I. Newton, and many other learn* 
ed perfons. 

Of Gregory* s Exercitationes Geometrical 

Jn the fame year 1668 came out Mr. James Gregory's Exercita^ 
^nes Geometrica^ in which are contained 

1, Appendicula ad veram circuli et hyperbolas quadraturam : 

2, N. Mercatoris quadratura hyperbolae geometrice^ demonftrata : 

3, Analogia inter lineam meridianam planisphaerii nautici et tan- 
gentes artificiales geometrice demonftrata ; feu quod fecantium na« 
turalium additio emciat tangentes artificiales : 

4, Item, quot tangentium naturalium additio efficiat fecantes arti* 
ficiales : 

5, Quadratura conchoid is : 

6, Quadratura cifToidis : & 

7. Mctho^iL 



>. 



LO GARI T H MS. $5 

7, Methodus facilis et accurita componendi fecantes et tangen* 
tes artificiales. 

The firft of thefe pieces, or the Appendicular contains fome far^ 
ther extenfion and illuftration of his Fera circuit it hyperbola quadratu-- 
ra^ occafioned by the animadverfions made on that work by the fa- 
mous mathematician and philofopher 'Huygens. 

In the ^d is demonftrated geometrically the quadrature of the hyper« 
bola, by which he finds a fenes fimilar to Mercator's for the logarithm, 
or the hyperbolic fpace beyond the firft ordinate (BI,y^. pa. 92.) 
In like manner he finds anotner feries for the fpac^ at an equal diftance 
within that ordinate, Thefe two feries having all their terms alike, 
but all the figns of the one plus, and thofe of the other alternately 
plus and minus, by adding the two together, every other term is can* 
celled, and the double of the reft denotes the fum of both fpaces. He 
then applies thefe properties to the logarithms ; the conclufioa from 
all which may be thus briefly exprefied ; 

fincc if — f iA + i^— I ^ *^c. =: the log. of ii-> 
and A+ ijfi + ^jt^ + lA^ &c, = the^.og. of -i_, 

therefore lA+^A^+^A^ i-^A' &c.= the log. of -—j or of the ra- 
tio of X — A'toi+A. Which may be accounted Mr. James Gre* 
gory's method of making lo^rithms. 

The remainder of this little volume is chiefly employed about 
the nautical meridian, and the logarithmic tangents and fecants. 
It does not appear by whom, nor by what accident, was difcovered 
the analegy between a fcale of logarithmic tai^gents and Wright's 
protrafiion of the nautical meridian line, which confifted of the 
ifums of the fecants. It appears however to have been firft publiflied, 
and introduced into the prafiice of navigation, by Mr. Henry Bond, 
who riientions this« property in an edition of Norwood's Epitome of 
Navigation, printed about 1645; and ^ he again treats of it more 
fully in an edition of Gunter's works printed in 1653, where he 
teaches, from this property, to refolve all the cafes of Mercator's 
failing by the logarit)\mic tangents, independent of the table of 
meridional parts. Thi^ analogy had only been found to be nearly 
true by trials, but not demonftrated to be a mathematical property, 
^uch demonftration feems to have been firft difcovered by Mr. Ni* 
cholas Mercator, who, defirous of making the moft advantage of 
this and another concealed invention of nis in navigation, by a 
paper in the Philof. Tranf. for June 4, 16669 invites the public to 
enter into a wager with him on his ability- to prove the truth or 
falfehood of the fuppofed analogy. This mercenary propofal how- 
ever ieems not to have been taken up by any one, and Mercator re- 
ferved his demonftraption. The propofal however excited the atten- 
tion of mathematicians to the fubje^l itfelf, and a demonftration was 
pot long wanting. The firft was publiftied about two years after by 
Gregory in the trad now under ponfideration, and from thence and 

other 



96 



CONSTRUCTION OF 




other limilar prpperries here demonftraced, he (hews in the laft ar- 
ticle how the tables of logarithmic tangents and fecants may 
eafily be computed from the natural tangents and fecants. The 
fubftance of which is as follows : 

Let AI be the arc of a quadrant extended 
ill a right line, and let the figure AHI 
be compofed of the natural tangents of 
every arc from the point A ere<Sled perpen- 
dicular to AI at their refpe^ive points: 
let AP, PO, ON, NM, &c, be the very 
fmall equal parts into which the quadrant is 
divided, namely, each jV) or 7^ of a degree, 
draw PB, OC, ND, ME, &c j^rpendicular to 
Ah ' Then it is manifeft from what had been 
monftrated, that the figures ABP, AGO, &c are the artificial 
fecants of the arcs AP, AG, &c, putting o for the artifici* 
al radius. It is alfo manifeft that the reflangles BO, ON, DM, &c 
will be found from the multiplication of the fmall part AP of the 
quadrant by each natural tangent. But, he proceeds, there is a little 
more difficulty in meafuring the figures AbP, BOX, CDV, &c ;. 
for if the firft differences of the tangents be equal, AB, BC» CD, 
&c will not differ from right lines, and then the figures ABP, BCX, 
CDV, &c will be right-angled triangles, and therefore any one, as 
HQG, will be = IQH x QG: but if the fecond differences be equal, 
thefaid figures will be portions of trilineal quadratices, for example 
HQG will be a portion of a trilineal quadratrix, whofe axis is pa- 
rallel to Qfl ; and each of the laft differences being Z, it will be 
QHG = fQHxQG-TTZxQG: and if the3ddifFcrenccsbeequal, 
the faid figures will be portions of trilineal cubices, and then (ha ll 
QHG be equal |QH x QG — V^^QBx Z x QG*— -J-- Z» x Qp* : 

when the 4th differences are equal, the faid figures are portions of 
trilineal quadrato-quadratices, and the 4th differences are equal to 
24 times the 4th power of QG divided by the cube of the latut 
re(3um \, alfo when the 5th differences are equal, the faid figures 
are portions of trilineal furfolids, and the 5th differences are equal 
to 120 times the furfolid of Qp divided h'y the 4th power of the 
latus re£tum ; and fo on in infinitum. What has been here faid of 
the compofition of artificial fecants from the natural tangents, it is 
remarked, may in like manner be underftood of the compofition of 
artificial tangents from the natural fecants, according to what was 
before demonftrated. It is alfo obferved that the artificial tangents 
and fecants are computed, as above, on the fuppofition that o is tho 
logarithm of i, and zoooooooooooo the radius, and 
2302535092994045624017870 the logarithm of 10 ; but that they 
may be more eafily computed, namely by addition only, by putting 
^j. of a degree — QG z= AP zr i , and the logarithm of ic =: 791570 
4467897819; for by this means |QH x QG isrr IQH r= QHG, 
and iQH x Qp -t^Z x QG ^iQVL -,VZJ= QHG, 

alfo 



LOOAtLtTHMS* 97 



itlTb fQH X QG-v/^«iQHxZxQG»— -i. Z» x Q^- fQH — 

v^yVQHxZ--J--Z- = QHG : And finally by one divifion only 

are found the artificial tangents and fecants to iooooooqcxx>ocxx)o 
the logarithm of 10, patting ftill i for radius, which are the differ- 
cnces of the artificial tangents and fecants in the table from that 
artificial radius ; and to make the operations eafier in multiplying by 
the number 7915704467897819, or logarithm of 10, a table is fet 
down of its produds by the firfl 9 figures* But if AP or QG be 
= 7^7 of a degree, the artificial tangents and fecants will anfwer to 
13 192840779829703 as the logarithm of 10, whofe firfl 9 multi- 
ples are alfo placed in the table. But to reprefent the numbers by 
the artificial radius rather than by the logarithm of io> the author 
direds to add ciphers, &c.— And fo much for Gregory^ Exercita^ 
iUnes Geometric^. 

The fame analogy between the logarithmic tangents and the 
meridian line, as alfo other fimilar properties, were afterwards more 
elegantly demonflrated by Dr. Halley in the Philof. Tranf. for Feb, 
1696, and various methods given f6r computing the fam/e, by exa- 
mining the nature of the fpirals into which the rhumbs are trani^ 
formed in the ftereographical projedion of the fphere on the plane 
of the equator : the doftrine of whicl> was rendered flill more eafy 
and elegant by the ingenious Mr. Cotes in his Logometria^ firft 
printed m the Philof. Tranf, for 17 14, and afterwards in the collec- 
tion of his works publifhed in 1732 by his coufm Dn Robert Smith,, 
who fucceeded him in the Plumian profefTorfhip of philofophy in 
the Univcrfity of Cambridge. 

The learned Dr Ifaac Barrow alfo, in his Lt^iones GeomitrM^ 
Le£f. XI. Jppend. firft publifhed in 1672, delivers a fimilar property, 
namely, that the fum of all the fecants of any arc, is analogous to 
the logarithm of the ratio of r+ s to r— j, or radius plus fine to radius 
minus fine; or, which is the fame thing, that the meridional parts 
anfwering to any degree of latitude, are as the logarithms of the ra*« 
tios of the verfed fines of the diftances from the two poles. 

Mr. Gregory's method for making logarithms was farther exem- 
plified in numbers, in a fmall tra£t 09 this fubje6t, printed in 1688, 
by one Euclid Speidell, a fimpk and illiterate perfon, and ion of 
John Speidell before mentioned among the firft writers on loga- 
rithms. 

Mr. Gregory alfo invented many other infinite feries, and among 
them thefe here following, viz. <? oeing an arc, / its tangent, and $ 
the fccant, to the radius r ; then is 



' "" ^ ■*■ ^r "*" i4r» "*■ ^29r^ '^ 8064 

o 




»8 CONSTRUCTION OF 

And if y and 9 be the artificial or logarithmic tangent and Tecant 
of the fame arc j, the whole quadrant being ;, and / =: atf -^ 
q \ then 

•*=''~6? "*■ i^"^ 504cr* ■*" 715761^ ^ • 

Alfo if /be the artificial fecant of 45^ and/H- / the artificial fecant 
of any arc 17, the artificial radius being o • then is 

The inveftigation of all which feries may be feen at pa. 298 a 
ffd. voU I. f>r Horfley*s learned and elegant commentary on Sir L 
Newton's works, as they were given in the Commercium Epiji&licum 
No XX without demonuration, and where the number 2 is alfo 
wanting in the denominator of the firft term of the feries expreffing 
the value of 0". 

Such then were the ways in which Mercator and Gregory ap- 
plied thefe their very fimple feries A — fA* + ; A* — -J A* Ice, and 
A + J A* + jA' + JA* &c, for the purpofe of computing logarithms. 
But tney might, as I apprehend, have applied them to this. purpofe 
in a fhorter and more dire6t manner, by computing, by their means, 
pnly a few logarithms of fmall ratios, in which the terms of the feries 
would have aecreafed by the powers of 10 or fome greater number, 
the numerators of all the terms being unity, and their denominators 
the powers of fo or fome greater number, and then employing thefe 
few logarithms, focomputed, to the finding of the logarithms of other 
and greater ratios by theeafy operations of mere addition and fub trac- 
tion. This might have been done for the logarithms of the ratios of 
the firft ten numbers, 2,3, 4, 5, 6, 7, 8, 9, 10, and 11, to i, in the 
following manner, communicated by Mr Baron Maferes.- In the 
firft place the logarithm of the ratio of 10 to 9, or of i to A, or of i 



to I -* JL, is equal to the f<|ries-rv -I ^ — A- -^ 

+ ' 5 X looooo *^- ^^ ^^^^ manner are eafily found the logarithms of 
the ratios of 11 to 10 j and then by the fame feries thofe of 121 to 
I20f and of 81 to 80, and of 2401 to 2400 j in all which cafes the 
feries would converge ftill fafter than in the two firft cafes. We may 
then proceed by mere addition and fubtra<aion of logarithms, as follows. 



Log.Y = L.fJ +L.V> 



L. 1 =L.V, 



L. f = L.i — L.|. 



L. ^= L.-\,^ + L.|, 
L.Vo^=L.Vo^-L,fa'lL. ^=aL.|, 
Having thus got the logarithm of the ratio of 2 to i, or, in com* 
mbn language, the logarithm of 2, the logarithms of all forts of 
even numbers may be derived from thofe of the odd numbers which 
are their coefficients with 2 or its powers. I would then proceed as 
follows. ^ L. 4 



LOGARITHMS. 99 



If. 4 :r z L* 1, I L. loo = 2 L* 10, 



L. 24or=rL.|$°^+L.2400, 



L, 7 =J L, 2401, 
L. 1 1 =: L V" + L. 9, 
L, 6 = L. s 4" L* 3. 



Lr. 9 £=; L. f 4" ^*49 L. 24 ^ L. 8 4- L. 3, 

L. 3 = I L. 9 lL.240O= L.ioo4-L(.24J 

Thus we have got the logarithms of 2, 3,4, 5, 6, 7, 8, 9, 10, and 1 1. 
And this is upou the whole, perhaps the beft method of .computing 
logarithms that can be taken. There have been indeed ibme me- 
thods difcovered by Dr. Halley, and other mathematicians, for com« 
puting the logarithms of the ratios of prime numbers to the next ad- ' 
jacent even numbers, that are ftilt (horter than the application of the 
foregoing lerics. But thofe methods are lefs fimple and eafy.to un» 
<(erftand and apply than thefe feries ; and the computation of loga« 
rithms by chefe feries, when the terms of them decreaie by the po« 
wers of 10, or of fome greater number, is fo very (bort and eafy 
(as we have feen in the iforegoing computations of the logarithms of 
the ratios pf .10 to 9, 11 to 10, 81 to 80, 121 to 120, &c,} that it is 
not worth while to feek for any ihorter methods of computing 
them. And this method of computing logarithms is very nearly the 
fame with that of Sir Ifaac Newton in his fecond letter to Mr» 
Oldenburg, dated October 1676, as will be feen in the following ar« 
tide. 

Of Sir Ifaac Newton^ s Methods. 

The excellent Sir I. Newton greatly improved the quadrature of 
rfie hyperbolical- afymptotic fpaces by infinite feries^ derived from 
the general quadrature of curves by his method of fluxions ; or rather 
indeed he invented that method himfelf, and the conftruAion of lo- 
garithms derived from it, in the year 1 665 or 1666, before the pub- 
lication of either Mercator's or Gregory's books, as appears by his 
letter to Mr. Oldenburg dated Oft. 24, 1676, printed in pa. 634 it 
ftq. vol. 3 of Wallis's works, and elfcwhere. The 
quadrature of the hyperbola, thence tranflated is to 
this effeft. Let dFj) be an hyperbola, whofe cen- 
ter 13 C, vertex F, and interpofed fquare CAFR .j« 
:r I* In CA take AB and Ab on each fide r: ^V 
or o' I : And, ercfting the perpendiculars ^\i^ bd ; ^-+- 
half the fum of the fpaces AD and Ad will be 

•• O I 4- °^^^' I O.OOOOI 1^ 0,0000001 tf_^ 

and the half diff. = 2^ + °J22£i. + 2:22°°fJ + 2f2°Sf22i &c. Which 
reduced will ftand thus, 

I.00OOOOOOO0O00 0.0050000000000 Thefumof tliefeo.i0536oji56577i8Ad. 
3333S33333 ajooooooo and the diflFcr. 0.0953101798043 i«AD» 

aooooooo 1666666 In like manner putting AB and Ab 

1428 J7 12506 each r: 0.2, there is obtained 

mi lOQ Ad ^0.223143551 3142, and 

^^ ^ ! AD:=:o.i8232i5567939. 

,0.1003353477310 0.0050251679267 

O 2 Haying 




too CONSTRUCTION OF 

Having thus the hyperbolic logarithms of the four decipiial num- 
bers o.8»o.9» I.I, and 1.2; and fince ^x^^:r2,ando.8 ando.9are 
lefs than unity ; add their logarithms to double the logarithm of i ^> 
and you will have 0.693147 1805597 the hyperbolic logarithm of 2. 

2x2x2 
To the triple of this add the logarithm of o.8, becaufe — ^ — == 

10, and you have 2.3025850929933 the logarithm of 10. Hence 
by one addition are found the logarithms of 9 and 11 : And thus 
the logarithms of all thefe prime numbers 2, 3, 5, 11 are prepared. 
Moreover, by only depreffing the numbers, above computed, lower 
in the decimal places, and adding, are obtained the logarithms of 
the decimals 0.98, 0.99, i.oi, i«02 ; as alfo of thefe 0.998, 0.9999 
X.OQi, 1.002: And hence by addition^nd fubtraftion will arife the 
logarithms of theprinies 7, 13, 17, 37, &c. All which logarithms 
being divided by the above logarithm of 10, give the common lo- 
garithms to be inferted in the table* 

And again a few pages farther on in the fame letter he refumes the 
conftrudion of the logarithms, thus : Having found, as above, the 
hyperbolic logarithms of 10, 0.98, 0.99, i.oi, 1.02, which* may be 
eneded in an hour or two, divide the lafl four logarithms by the loga- • 
rithm of 10, and adding the index 2, you will have the tabular loga- 
rithms of 98, 99, 160, loi, I02. Then by interpolating nine 
means between each of thefe, will be obtained the logarithms of all 
numbers between 980 and 1020 ; and agaiTi interpolating o means 
|>etween every two numbers from 980 to 1000, the table will be (a 
far conftruftcd. Then /rom thefe will be colleftcd the logarithms 
of all the primes under 100, together with thofe of their multiples : 
all which will require only addition and fubtradion ; for 

jQ 9984X1010 10 y98 _ ^ 99 looi ic» 

\/ 9945 =^»T-5>VT-7»7-">7y7r=i3.-6-=i7* 
43rU- '9)763^17- ^3* 7x17 -^9>^ 3i,-^ =37, - = 41, 

989 -« .^ 987^ _ ,22iL- CO -^l Cft J?i^* - ^ •9«49 _^^ 

— -43» T7 " 47> 11x17 ""^3' 13x13 ""^^^ix 81- ^'>7x4T'"^* 

994 _ ^- 99tg _ 9954 — ^^ 996 _ g' 9968 _ 9894 _ 

li"-^ 7^8x17-73^7x18- /9' 12 - ^3>7S<T6 -^9» 65ri7 = 97- 

This quadrature of the hyperbola, and its application to the con- 
flrudion of logarithms, are ftill farther explained by our celebrated 
author in his treatife on Fluxions, publifticd by Mr Colfon in 1736, 
where he gives all the three feries for the areas AD, Ad, Bd, in ge- 
neral terms, the former the fame as that publifhed by Mercator, and 
the latter by Gregory ; and he explains the manner of deriving the 
latter feries from the former, namely by uniting together the two 
feries for the fpaces on each fide of an ordinate, bounded by 
€ther ordinate^ at equal diftances, every 2d term af each feries is 

rancelled^ 



LOGARITHMS. xoi 

cancelled, and the rcfult is a feries converging much quicker than 
cither of the former. And, in this treatife on fluxions, as well as 
in the letter before quoted, he recommends this as the moft commo« 
dious method ofconftruairig a canon of logarithms, computing by 
the feries the hyperbolic fpaces anfwering to the prime numbers 
2, 3, 5, 7, II, &c, and dividing them by 2.3025850929940457, 
which is the area correfponding to the number 10, or elfe multiply- 
ing them by its reciprocal 0.4342944819032518, for the common lo- 
garithms. *' Then the logarithms of all the numbers in the canon 
which arc made by the multiplication of thefe, are to be found by 
the addition of their logarithms, as is ufual. And the void places are 
to be interpolated afterwards by the help of this theorem : Let « bca 
number to which a logarithm is to be adapted, x the difference be- 
tween that and the two neareft numbers equally diftant on each fide, 
whofe logarithms are already found, and let d be half the differ-? 
«nce of the logarithms j then the required logarithm of the number 

n jwill be obtained by adding d+~V^^ ^<^ «^ ^^^ logarithm of 
the kfs number," This theorem he demonftrates by the hyperbolic 
areas, and then proceeds thus s « The two firft terms ^+ — of this 

feries I think to be accurate enough for the conftruSion of a canon 
ofiogarithm8,.even though they were to be produced to 14 or 15 
jfieures ; provided the number whofe logarithm is to be found be not 
lefs than looo* And this can giye little trouble in the calculation, 
becaufe x is generally an unit, or the number 2, Yet it is not ne* 
ceflary to interpolate all the places by the help of this rule. For the 
logarithms of numbers which are produced by the multiplication or 
divifion of the number laft found, may be obtained by the numbers 
whofe logarithms were had before, by the addition or fubtradion of 
their logarithms. Moreover by the differences of the logarithms, and 
by their ad and 3d differences, if there be occafion, the void places 
may be more expeditioufly fupplied ; the foregoing rule being to be 
applied only when the continuation of fome full places is wanted, in 
order to obtain th'ofe differences, &c." So that Sir I. Newton of 
himfelf difcovered all the feries for the above quadrature which were 
found out, and afterwards publiihed, partly by Mercator and partly 
by Gregory ; and thefe we may here exhibit in one view all toge- 
ther and that in a general manner for any hyperbola, namely putt- 
ing CA =! tf, AF = *, and AB = Ab = jr ; then will 

PD -- «lL, and bd = -3- ; whence the area ^ 

2a 341* j{a^ £0* 

Ad=i* + — + - +- +_ &c. 

In 



164 CONSTRUCTION OF 

hyperbola, with a fpeedy methpd for finding the number from thtf 
given logarithm.'* 

Inftead of the more ordinary definition of logarithms, numerorum 
froportionalium aquidiffenntes comitesj in this trad our learned author 
adopts this oth^r, numeri rathnum txponenteSf as being better adapted 
to the principle on which logarithms are here conftruded, where 
thofe quantities are not cojnfidered as the logarithms of the numbers, 
for example, of 2, or of 3, or of 10, but as the logarithms of the ra- 
tios of I to 2, or I to 3, or i to lo. In this confideration he firft 
purfues the idea of Kepler and Mercator, remarking that any fuch 
ratio is proportional to, and is meafured by, the number of eaual 
ratiunculae contained in each ^ which ratiunculae are to be under- 
flood as in a continued fcale otproportionals) infinite in number, be- 
tween the two terms of the ratio ; which infinite number of meaii 
proportionals is to that infinite number of the like and equal ratiun- 
culae between any other two terms, as the logarithm of the one ra- 
tio is to the logarithm of the other : thus, if there be fuppofed be- 
tween I and 10 an infinite fcale of meao proportionals^ whofe num- 
ber is lOOOOO &c in infinitum ; then between i and 2 there will be 
30102 &c of fuch proportionals ; and between i and 3 there will be 
47712 &c of them ; which numbers therefore are the logarithms of 
the ratios of i to 10, i to 2, and i to 3. But for the fake of b!s 
mode of conftruding logarithms, he changes this idea of equal rati- 
unculae for that of other ratiunculae, fo conftituted as that the fame 
infinite number of them fhall be contained in the ratio of i to every 
other number whatever ; and that therefore thefe latter ratiunculae 
will be of unequal or different magnitudes in all the diflFerent ratios, 
and in fuch fort that, in any one ratio, the magnitude of each of the 
ratiunculae in this latter cafe, will be as the number of them in the for- 
mer. And therefore if between i and any number propofed, there be 
taken any infinity of mean proportionals, the infinitely fmall augment 
or decrement of the firft of thofe means from the firft term i, will be a 
ratiuncula of the ratio of x to the faid number; and as the number of 
all the ratiunculae in thefe continued proportionals is the fame, their 
fum or the whole ratio will be diredly proportional to the magnitude 
of one of the faid ratiunculae in each ratio. But it is alfo evident 
that the firft of any number of means between i and any number, is 
always equal to fuch root of that number whofe index is expreifed by 
the number of thofe proportionals from i ;»fo if m denote the number 
of proportionals from i, then the firft term after i will be the mth 
root of that number. Hence the Indefinite root of any number being 
extra(Sled, the differentiola of the faid root from unity, fhall be as the 
logarithm of that number. So if there be required the logarithm of the 

ratio of I to 1 + y i the firft term after i will be T+7^, and there* 
fore the required logarithm will be as r4rj)*-i. But, by thebino. 
mlal theorem, rx^* is := 1 + •^^ + ^« ^-^q* + — •i^.il-!!i^i 
Uc } or by omitting the i in the compound numerators, as infi- 

nitclj 



LOGARITHMS. le; 

nltely fmall in refpeA of the infinire nutabcrtity the fame ferieswill' 

become i + hl + ll^ f +i-^'^f^'^*°^ ^T abbreviation 

it is I + '■a — — a^ +— ^ — -9* &C} and hence finding the 

differentiola by fubftra£ling i, the logarithm of. the ratio of i to i 
+ qwinhezs^intof — iq^ + i^^ — ii^ ^ \q^ —if tuc. Now 

the index m may be taken equal to any infinite number, and thus 
all the varieties of fcalesof logarithms may be produced : lb, if i» be 
taken loooocx) &€» the theorem will give Napier's logarithms ; but if 
m be taken equal to 230258 &c, there will arife Briggs's logarithms. 
This theorem being for the increafing ratio of i to i + ^ » if that 
for the decreafing ratio of I to i — ^ be alfo fought, it will be ob- 
tained by a proper change of the figns, by which the decrement of 
the firft of the infinite number of proportionals will be found to be 
^ into y + ff* + i^* + ij^ &c, which therefore is as the logarithm 

of the ratio of i to i — ^. 

Hence the terms of any ratio being a and by q becomes Jll^ or the 

difference divided by the lefs term, when it is anincreafing ratio; or 
q =: -^ when the ratio is decreafing, or as ^ to a. Wherefore the 

logarithm of the fame ratio may be doubly exprefled ; for putting x 
for the diiterence b^a oi the terms, it will be 

either - into -——-+-—— — - &c, 
m 4» 2tf* id} A^"* ' 

1 • X ** 9^ x^ 

But if the ratio of j to ^ be fuppofed divided into two parts, namely, 
into the ratio of a iiy\a+ {b or fa;, and the ratio of ^z to by then 
will the fum of the logarithms of thofe two ratios be the logarithm 
of the ratio of a to b. Now by fubfiituting in the foregoing feries^ 
the logarithms of thofe two ratios will 

be— into — | r-K— r-H rH 7 ^ 

I X Jt* X^ X^ x^ 

and —into—— — -H r — — 7+ — r^c; and fo the fum 

I 2X 2Jt' 2x^ 2sp 

— into — ^""i + rs'^" "7 &c willbethelog/oftheratiooftftoi^. 

Moreover, if from the logarithm of the ratio of ^ to \% be taken 
that of \% to by we fhall have the logarithm of the ratio of ab to |z^ ; 
and the half of this gives that of '/ab to |z, or of the geometrical 
mean to the arithmetical mean. And confequently the logarithm of 
this ratio will be^ual to half the difference of that of the above two 

ratios, and will therefore be i into — » + --3 + t-x + 5^ &c. 

2% /^Xt ux ox 

The above icries are fimilar to fome that were before given hj 
Newton and Gregory for the fame purpofe, tieduced from the confi*- 
deration of the hyperbola. But the rule which is properly our au» 

P tbor's 



,o6 CONSTkUCTIONOF 

thor*s own, is that which follovvs, and h derived from the fericd 
above given for the logarithm of the fum of two ratios. For the 
ratio of fli to iz* or Ja* +^jfi+ "4^% having the difFcrence of its 
terms Ja* — i^^ + i** or \d — i^ * or J**, which in the cafe of find- 
ing the logarithms of prime numbers is always i, if we call the funi 
of the terms Jz* 4- ^* = /, the logarithm of the ratio of ^ab to 
la + ib or i« will be found to be 

I.I 1 I.I I « 

- into - -J 5 + —[b + "TTi + — Th ^^• 

And thefc rules our learned author exemplifies by fome cafes in 
numbers, to Ihew the eafieft mode of application in praftice. 

Again by means of the fame binomial theorem he refolvcs with 
equal facility the revcrfe of the problem, namely, from the log- 
arithm given, to find its number or ratio : For as the logarithm 

of the ratio of i to i + ^ was proved to be i +^>«-^ i, and that 

of the ratio of i to i — ^ to be . . • i — i^* i hence, 
calling the given logarithm L, in the former cafe 

it will be X +^^w zi i + L, 

and in the latter 7^-^* = i — i^ ; 

and therefore i +^ =T+i:«\ ^^^^ .^ ^^ ^^^ ^^^^.^^ ^^^^^^^ 

aod 1— ^=: I — L^^J 

i+q=:t + mL + |»i^L*+ im^L^ + ^WL*+ Th>«* Ls &c, 

andi — y=i— mL + |«*L*-. Jm' L' + V^w* L* — tIo'w^ Ls &c. 

HI being any infinite index whatever, diflfering according to the 

fcale of logarithms, being looo &c in Napier's or the hyperbolic 

loearithms, and 2302585 &c in Briggs*s. 

If one term of the ratio, of which L is the logarithm, be given, 
the other term will be eafily obtained by the fame rule : For if 
L be Napier's logarithm of the ratio of a the lefs term to b the 
greater, then according 2is a or b is given we {hall have 
b = a into 1; + L + f L* + ^L' + ,\IA &c, 
a=:b into i — L + |L» — ^U + ^\IA &c. 

Whence, by the help of the logarithms contained in the tables, may 
eafily be found the number to any given logarithm to a great extent : 
For if the fmall difference between the given logarithm L and the 
neareft tabular logarithm, either greater or lefs, be called /, and the 
number anfwering- to the tabular logarithm a when it is lefs than 
the given logarithm, but b when greater; it will follow that the 
number anfwering to the logarithm L will be either 

a into i + l + il* +iP + ,1^/4 + ^j^A &c, 
or b into i—l+il^ — iP + ^'^/4 — ^ j^/s &c. 
which feries converge fo quick, / being always very fmall, that the 
firft two terms i ±/ are generally fufficient to find the number to 
10 places of figures. 

Dr. Halley fubjoins alfo an eafy approximation for thefe feries, 
by which it appears that the number anfwering to the log. is nearly 



LOGARITHMS. to; 



t+l/ 1-1/ 



Mrhere » is = 434294481903 &c r: -. 

Of Mr. Sharp's Methods. 

The labours of Mr. Abraham Sharp, of Little- Horton near Brad- 
ford in Yorkfliire, in this branch of mathematics, were very great and 
meritorious. His merit however confilted rather in the improvement 
and illuftration of the methods of former writers, than in the inven- 
tion of any new ones of his own. In this way he greatly extended 
and improved Dr. Halley's method above defcribed, as alfo thofe of 
Mercator and Wall is ; illuftrating chefe improvements by extenflve 
calculations, and by them computing our table 5, 'confining of the 
logarithms of all numbers to 100, and of all prime numbers to iioo> 
each to 61 places. He alio compofed a neat compendium of the beft 
methods for computing the natural fmes, tangents, and fecants, 
chiefly from the rules before given by Newton : and by Newton's or 
Gregory's feries a:=-t — ^i^+\t^ — 7/^ &c, for the arc in terms of 
the tangent, he computed the circumference of the circle to 72 
places, namely from the arc of 30 degrees, whole tangent / is rr i/\ 
to the radius i. Other aftonifhing inflances of his induftry and la- 
bour appear in his Geometry Imprcrd'd^ printed in 1717, and figned 
A. S* Phtlomathy from whence the faid table of Io|;arithms was ex- 
tracted. This ingenious man was fometime afliirant at the Royal 
Obfervatory to Mr. Flamfteed the firft Aftronomer Royal j and, 
being one of the moft accurate and indefatigable computers that ever 
exifted, he was for many years the common refource for Mr. Flam- 
fteed, Sir Jonas Moore, Dr. Halley, &c, in all intricate and trou- 
blefome calculations. He afterwards retired to his native place at 
Little- Horton, where, after a life fpent in intenfe ftudy and calcu- 
lations, he died the i8th of July 1742, in the 91ft year of his age. 

Of the ConftruSIion of Logarithms by Fluxions* 

It appears by the very definition and defer iption given by Napier 
of his logarithms, as ftated in page 42 of this introduction, that the 
fluxion of his, or the hyperbolic logarithm, of any number, is a fourth 
proportional to that number, its logarithm, and unity; or, which is 
the fame, that it is ^qual to the fluxion of the number divided by the 
number : For the description (hews that zi : z<2 or i : : ^i the flux- 

ion of zl : ka^ wfeich therefore is = — ; but i^ is alfo equal to the 
fluxion of the logarithm A &c by the defcription ; therefore the flux- 
ion of the logarithm is equal to ^, the fluxion of the quantity di- 
vided by the quantity itfelf. The fame thing appears again at art. 
2 of that little piece in the appendix to his Co»Jiru^io Logarithmo* 
fum^ entituled Habitudines Logarithmorum isf ftiorum naturalium »«- 

P 2 merorum 



stf CONSTRUCTION OF 

mer9rum Invtam^ where he obferves that, as any greater quantity is 
to a lefs, fo is the velocity of the increment or decrement of the log- 
arithms at the place of the lefs quantity, to that at the greater. Now 
this velocity of the increment or decrement of the logarithms being 
the fame thing as their fluxions, that proportion is this x : a : i flux, 
log. a : flux. log. x ; hence if ^7 be = i, as at the beginning of the table 
of numbers, where the fluxion of the logs^ is the index or charade- 
riftic r, which is alfo i in Napier's or the hyperbolic logarithms, and 
43429 ^c in Briggs's, the fame proportion becomes x : i i \c \ flux. 
log. X ; but the conftant fluxion of the numbers is alfo i, and there- 

fore that proportion is alfo this x i x :: c '.^ = the fluxion of the 

logarithm of jr; and in the hyperbolic logarithms, where r is = i, it 

becomes — =r the fluxion of Napier's or the hyperbolic logarithm of 

*. This fame property has alfo been noticed by many other authors 
fince Napier's time. And the lame or a fimilar property is evidently 
true in all the fyftems of logarithms whatever, namely that the mo- 
dulus of the fyftem is to any number, as the fluxion of its logarithm is 
to the fluxion of the number. 

Now from this property, by means of the dodrine of fluxions, 
are derived other ways for making logarithms, which have been il- 
luflrated by many writers on this branch, as Craig, Jo. Bernoulli, 
and almoft all the writers on fluxions. And this method chiefly 
confifts in expanding the reciprocal of the given quantity in an in- 
finite feries, then multiplying each' term by the fluxion of the faid 
quantity, and laflly taking the fluents of the terms ; by which there 
arifes an infinite feries of terms for the logarithm fought. So, to 
find the logarithm of any number N s put any compound quantity 

for N, as fuppofe '' ; 
then the flux, of the log. or 2 being ' -4— = --— -j-^— "^fec, 
the fluents give log. of N or log. of 1±-^ = - — ff + ^— ^ &c. 
And writing ^ ;c for x gives log. • = — ^ ^ &c. 

® ^ ^ n n 2n* ^fi^ 4ii* 

Alfo becaufe — ~— = i -r * , or log. —^ = o — log. — =^ 

we have log. —. — = — -i ;^ &c. 

and log. — ^ = + -+ 4+ 4+-II&C 

And by adding and fubtrading any of thefe feries to or from one 
another, and multiplying or dividing their correfponding numbers, 
various other ieries for logarithms may be found, converging much 
quicker than thefe do. 

In 



LOGARITHMS. 109 

In like manner by afliiming quantities otherwife compounded for 
the value of N, various other forms of logarithmic feries may be 
found by the fame means. 

Of Mr^ Cotes* s Logomttria. 

Mr. Roger Cotes was elefted the firft Plumian profeflbr of aftro* 
nomy and experimental philofophy in the univerfity of Cambridge^ 
January 1706^ which appointment he filled with thegreateft credit^ 
tillhedied the5thof June 1716, in the prime of 11 fe, having not quite 
compleated the 34th year of his age. His early death was a great iofs 
to the mathematical worlds as his eenius and abilities were of the 
brighteft oider^ as is manifcft by the fpecimens of his performance 
given to the public. Among thefe are his Logometria^ iirft printed in 
number 338 of the Philofophical Tranfadions, and afterwards in his 
Harmonia Minfurarum publifhed in 1722 with his other works, by 
his coufin and fucceflbr, in the Plumian profefTorfliip, Dr. Robert 
Smith, (n this piece he firft treats in a general way of meafures of 
ratios, which meafures he obferves are quantities of any kind whole 
magnitudes are analogous to the magnitudes of the ratios^ thefe 
magnitudes mutually increasing and decreafing together in the fame 
proportion. He remarks that the ratio of equality has no magni» 
tude, becaufe it produces no change \^y adding and fubtra6ling ; that 
the ratios of greater and lefs inequality, are of different affe^ions; 
and therefore if the meafure of the one of thefe be confidered aspo- 
fitive, that of the other will be negative; and the meafure of the ra- 
tio of equality nothing : That there are endlefs iyftems of thefe 
which have all their meafures of the fame ratios proportional to 
certain given quantities, called moduli^ which he defines after- 
wards, and the ratio of which they are the meafures, each in its pe- 
culiar fyftem, is called the modular ratio, ratio modularise which ra- , 
tio is the fame in all fyftems. He then adverts to logarithms, which 
he confiders as the numerical meafures of ratios, and he defcribes the 
method of arranging them in tables, with the ufe of them in multi- 
plication and divifion, raifing of powers and extra£ling of foots, by 
means of the correfponding operations of addition and fubtra<3ion, 
multiplication and divifion. 

After this introdu&ion, which is only a ilight abridgment of the 
doi^ine long before very amply treated of by others, and particu- 
larly by Kepler and Mercator, we arrive at the firft propofition, 
which has juftly been cenfured as obfcure and imperfed, feemingly 
through an affeftation of brevity, intricacy, and originality withouft 
fufficient rooni for a difplay of this qualification. The reafoning m 
this propofition, fuch as it is, f.'ems to be ibmething betWlben that of 
Kepler and the principles of fluxions, to which the quantities and 
cxpreffions are nearly allied. However as it is my duty rather to 
narrate than explain, I fhall here exhibit it exadly as it ftands. This 
propofition is to determine the meafure of any ratio, as for inftance 
that of AC to AB, and which is cffeded in this manner : Conceive 

the 



im 



no CONSTRUCTION OF 

the difference BC to be divided •—! ijj a 

into innumerable very fmall par- ^ » " Q^ C 

tides as PQ^, and the ratio between AC and AB into as many fucb 
very fmall ratios, as between AQ^and AP : then if the magnitude of 
the ratio between AQ^and AP be gfven, by dividing there will alfo 
be given, that of PQ^to AP ; and therefore, this being given, the 
magnitude of the ratio between AQ^and AP may be expounded by 
the given quantity ^; for, AP remaining conftant, conceive the 

particle PQ.to be augmented or diminiflied in any proportion^ and 
in the fame proportion will the magnitude of the ratio between AQ^ 
and AP be augmented or diminrfhcd : Alfo, taking any determinate 

quantity M, the fame may be expounded by M x jjp ; and there- 
fore the quantity M x ?2.will be the meafure of the ratio between 

AQ^and AP. And this meafure will have divers magnitudes, and be 
accomodated to divers fyftems, according to the divers magnitudes of 
the aflumed quantity M, which therefore is called the modulus of the 
fyftem. Now like as the fum of all the ratios AQ^to AP is equal to 
the propofed ratio AC to AB, fo the fum of all the meafures M x 

??- (found by the known methods) will be equal to the required 

ineafure of the faid propoled ratio. 

The general folution being thus difpatched, from the general cx- 

{»reffion he next deduces other forms of the meafure in feveral corol- 
aries and fcholia: as ift. The terms AP, AQ^, approach the 
liearer to equality as the fmall difference PQ^is lefs ; fo that either 

M X Tp or M X ^will be the mealure of the ratio between AQ^ 

and AP to the modulus M. 2d, That hence the modulus M is to the 
meafure of the ratio between AQ^and AP, as either AP or AQ^ is to 
their difference PQ. 3d, The ratio between AC and AB being 

given, the fum of all the tI'W*^^ ^ given; and the fum of all the 

A" 

M x' ^'s as M .• therefore the meafure of any given ratio is as the 

modulus of the fyftem from which it is taken. 4th, Therefore, ia 
every fyftem of meafures, the modulus will always be equal to the 
meafure of a certain determinate and immutable iratio ; which there- 
fore he calls the modular ratio. 5th, To illuflrate the folution by an 
example: let z be any determinate and permanent quantity, jr a va- 
riable or indeterminate quantity, and x its fluxion ; then to find the 
meafure of the ratio between z + x and z — jf, put this ratio equal 
to the ratio between y and i, expounding the number^ by AP, its 
fluxion^ by PQ, and i by AB : then the fluxion of the required 

y ' 

meafure of the ratio between jf and 1 is M x -. Now for jr reftore its 

val.-—^—, and for J the flux, of that val. , fo ihall the flux, qf 

the 



• 



LOG ARI T H MS. »ti 

the mfcafure become 2 M x — ^^ or a M into - + ^ + ^ &c* 

, zz ^ XX a K» St* 

And therefore that meafure will be 2 M into- + — r+ T~c ^^* 

z 3«* s* 

In like manner the meafure of the ratio between i + v and i 

will be found to be - - - M into v — fv^+|v^-|t;* &€• 

And hence, to find the number from the logarithm given, be reverts 

the feries in this manner: if the laft meafure be called m, we 

fhall have ^orQ^iz v*-|v* + ^^'— |«'* + 4^* &c^ 
and therefore Q^= . • . v* — v^+iiv^ — :^v^ &c, 

and Q3= t;3— i v*+iv^icc^ 

and Q^zr .,.•••••• v* — 2^&c> 

and Qi= v* &c; 

then by adding continually we fhall have 

Q.+ iQ!+.iQi = v— At/* + ^'uv*&c, 

a+ la* + iO! + tvQ! + ts^Q! = ^ &c, 

that IS v = Cl+ I Q! + 4Qf + f^O: + rr^y Q! &c. And therefore 
the required ratio of i + v to i is equal to the ratio of i + Q^-j- 
i Qi&c to I. Put now /» = M, or Q^n i, and the above will be- 
come the ratio of i + i + J + J + t';: + rh &c to i for the con- 
ftant modular ratio. In like manner, if the ratio between i and 
I — f; be propofed, the meafure of this ratio will come out M into 

V + ii/^ +T^' +|«^&C} which being called my and^ — Q, 

that ratio will be the ratio of i to i - Qjf i Q^— 4 Q^ + ^i, Qj&c* 
And hence, taking m = M, or Q^ = i, the faid modular ratio 
will alfo be the ratio of i toi-i- + i-^ + ^\^ -ri^ &c- And the 
former of thefe expreffions for the modular ratio comes out the 
ratio of 2,718281828459 &c to i, and the latter the ratio of x 
to 0,367879441171 &c. 

In the 2d prop, our learned author gives dire£lions for con- 
ftruding Briggs's canon of logarithms, namely, firft by the general 

feries 2M into 1 r + — . &c finding the logarithms of a few 

z • 32J 5* 
fuch ratios as that of 126 to 125, 225 to 224, 2401 to 2400, 4375 to 
4374, &c, from whence the logarithm of 10 will be found to be 
2,302585092994 &c, when M is i ; but fince Briggs's log. of 10 is 

1, therefore as 2,302585 &c is to the modulus i, fo is I (Briggs's log. 
of 10) to 0,4^4294481903 &c, which therefore is the modulus of 
Briggs's logarithms. Hence he deduces the logarithms of 7, 5, 3, and 

2. In like manner are the logarithms of other prime numbers to be 
found, and from them the logarithms of compofite numbers by addi» 
tion find fubtniftion only • . . 

Ho 



it* CONSTRUCTIOtT OF 

He then remarks that the firft term of the general feries 2M into 
1 1 H . &c will be fufficient for the fogarithms of intermc- 

diate numbers between thofe in the table, or even for numbers be« 
yond the limits of the table. Thus, to find the logarithm anfwering 
to any intermediate number; let a and e be two numbers, the one 
the given number, and the other the neareft tabular number, a being 
the greatecand e the lefs of them ; put z:^ a + e their fum, ^r := 
a — i their difFerence, a =r the logarithm of the ratio of a to e^ 
that is the excefs of the logarithm of a above that of ^ : ib fhall the 

faid difference of their logarithms be ^ zr aM x — very nearly.-— 

And if there be required the number anfwering to any given inter- 
mediate logarithm, becaufe x is =z 

a M* 2 M^ 2 Mx - - Ais xe ' . 

~ or i — , thercf. x =r .^ or,rjr — t- very neanv* 

In the 3d prop, our ingenious author teaches how to convert the 
canon of logarithms to logarithms of any other fyftem, by means of 
their moduli. ^ And in feveral more propofitions he exemplifies the 
canon of logarithms in the foiution of various important problems 
in geometry and phyfics ; fuch as the quadrature oF the hyperbola, 
the defcription of the logiftica, the equi-angular fpiral, the nauti- 
cal meridian, &c ; the defcent of bodies in refifting mediums, the 
denfity of the atmofphere at any altitude, &c, &c* ^ 

Of Do^or Taylor^ s ConJIruSfion of Logarithms* 

m 

Dr. Brook Taylor, (a very learned mathematician, ftcretary to 
the Royal Society, and who died at Somerfet-houfe, Nov. 1731) gave 
the following method of conftruf^ing logarithms in number 352 of 
the Philofophical Tranfaffcions. His method is founded on thefe 
three confiderations : ift, that the fum of the logarithms of any two 
numbers is the logarithm of the produ£l of thofe numbers ; ad, that 
the logarithm of i is nothing, and confequently that the nearer any 
number is to i, the nearer will its logarithm be to o ; 3d, that the 
produdl of two numbers or factors, of which the one is greater and 
the other lefs than i, is nearer to i than that fadtor is which is on 
the fame fide of i with itfelf ; fo of the two numbers ^ and ^, th& 
produd § is lefs than i» but yet nearer to it than | is, which is al(b 
lefs than i* On thefe principles he founds the prefent approxima* 
tion, which he explains by the following example. To find the 
relation between the logarithms of 2 and 10: In order to this he 

affumes two fra^ons as .and — ^1 or —rand — , whofe numera* 

100 10 10^ 10 

tors are powers of 2*, and their denominators powers of 10, the one 

fradion being greater and the other lefs than unity or i. Having 

fet theft two down^ in the form 4^ decimal fradjoas] below each 

other 



LOGARITHMS. 



"5 



diher in the lirft column of the following table, and In the fecond 
column A and B for their logarithms, expreffingby an equation how 
they are compofed of the logarithms of 2 and 10, the numbers in 
queftion, thofc logarithms being denoted thus, / 2 and / 10. Then 



i^iSoooooooooo A 

0,800000000000 B 

1,024000000000 C 

10,9905^2031429 

1,004336277664 
0,998959536107 
1,00016289416$ 

0,999936*8^874 
1,00003544^1115 

0,99997 '7^0830 

1,000007161046 

10,999993^03514 

1,000000304511 



o. 



,999999764687 



D: 

£: 
F: 

G: 
H = 
I = 
K= 
L: 

M: 
N: 

O: 



A + 
B + 

:C + 

D + 

:E+ 
F-- 

G-- 

H + 
I -- 
K-- 

L + 



• •— 


7/1. 


• •» 


3/* 


B = 


10/2 


90 = 


03/2 


aD = 


169/2 


3E = 


48C/2 


4F = 


2I36/2- 


«G- 


I330I/2 


iH = 


28738/2 


I = 


42039/2 


K = 


70777/2 


^b= 


254370/2 


M= 


525147/2 


8N=< 


H070I6/2 



2/101/2 

/lO 

5/10 

28/10 

59/10 

146/10 

643/10 

4004/10 

8651/10 

12655/10 

• 21306/10 

76573/10 

97879/10 

1838335/10 



comp.ar.2353i3 

0=36451104- a3's3i3N^23oa5858a5i8772—693i47400$72/io 



^ ■■ n i l 

>o,28 I 

<o,33 

> 0,300 

< 0,30107 

> 0,301020 

< 0,3010309 

> 0,30102996 

< 0,301029997 

> 0,3010299951 

< 0,3010299950 

> 0,30102999502 

< 0,30102999567 

> 0,3010299956635 

< 0,3010299956640 

> 0,301029995663982 



multiplying the two n^imbers in the firft column together, there is 
produced a third number 1,024, ^g^i^ft which is written C, for its 
logarithm, expreffing likewife by an equation in what manner C 
is formed of the foregoing logarithms A and B. And in the fame 
manner the calculation is continued throughout; only obferving 
this compendium, that before multi|;>]ying the two laft numbers 
already entered in the table, to confidcr what power pf one of thein 
muft be ufed to bring the produ& the neareft that can be to unity. 
Now, after having continued the table a little way, this is found 
by only dividing tne differences of the numbers from unity one by 
the other, and taking the neareft quotient for the index of the power 
fought. l*hus, the fecond and third numbers in the table being 
0,8 and 1,024, their differences from unity are 0,200 and 0,024; 
hence 0,200 -r 0,024 gives 9 for the index ; and therefore malti* 
plying the 9th power of 1,024 ^Y o>^ produces the next number 
09990352031420, whofe logarithm is D = B + 9 C. 

When the calculation is continued in this manner till the num* 
bers become fmall enough, or near enough to i, the laft logarithm 
Js fuppofed equal to nothing, which gives an equation expreffing 
the relation of the logarithms, and from whence the required log- 
arithm is determined. Thus, fuppofmg G = o, we have 
2136/2 — 643/10 = o, and hence, bec^ufe the logarithm of 10 la 

I, we obtain /a = 77^ =:0>30i02996, too fmall in the laft figure; 

which fo happena becaufe the number correfponding to G is greater 
than I. And in this manner are all the numbers in the third or 
lafl column obtained^ which are continual approximations to tho 
logarithnis of 9. 
^ Q, Ther^ 



114 



CONSTRUCTION OF 



Th.ere is another expedient which renders this calculation ftiil 
ftnrte r, and it is founded on this confideration ; that wllisn x is fmall, 
1 +*" is nearly =: i + »jr. Hence if i + *• and i — z be the 
two laft numbers already found in the firft column of the table, the 
produd of their powers 7+1?" x i — i*^ will be nearly ;= i ; and 
hence the re lation of m and n may be thus found, iT+l?* x i — »" 
is nearly =: i + /«jr x i — «» zr i + w-r — «z — »i« jr% = i -l-w-r — »» 
nearly , w hich being alfo = i nearly^ therefore m: n\\%i x-.i 
Ij'-x: L I +x ; whence *• A i^« + z /. i +x = o. For example, 
let 1,024 ^^^ 0>99^35^ ^^ ^he laft numbers in the table, their log- 
arithms being CandD: here we have 1,024 rr i + ;if, and 
0,990352 =1 — z; confequently x = 0,024, and z =: 0,009^48 ; 

and hence the ratio - in fmall numbers is — . So that for finding: 

the logarithms propofcd, we may take 500 D + 201 C =: 48510 / 2 
— 14603/10 = o; which gives /a .•= o,30i0307. And in this man- 
ner are found the numbers in the laft line of the table. 

Of Mr. Long's Method. 
In number 339 of the Philofbphical Tranfaflion?, are given a 
brief table and' method for finding the logarithm to any number, 
and the number to any logarithm, by Mr. John Long, B. D. Fel- 
low of C. C. C. Oxon. This table and method are fimilar to thofe 
defcribed in chap. 14 of Brij;gs's jfriih. Logar. differing only in 
this, that in this table by Mr. Long, the logarithms^ in each clafs, 
are in arithmetical progref&on, the common difference being i ; 
but in Briggs's little table the column of natural numbers has the 
like common -difference. The table confifls of eight claifes of 
logarithms and their correfponding numbers as follows : 



Lo. 



9 
,8 

,6 

A 



Nat. Numb. 

7794^82347 

6,30957344*; 
5,011872336 

3,981071706 

3,162277660 

2,51188643a 

1,995262315 

1,584893193 

1,258925412 



►09 
8 

7 
6 



4 

3 

2 

1 



1,230268771 
1,202264435 

I1' 74897555 
1,148153621 

5|i^ 1 22018454 

1,096478196 

1,071510305 

1,047128548 

1,023292992 



Log. jNat. Numb.M Log. iNat. Numb. 



,00 



7 
6 

5 

3 

2 

I 



,0009 
8 

7 
6 

5 

4 

3 

2 

1 



1,0209^9484 
1,018591388 
1,016248694 
1,013911386 

1,011579454 
1,009252886 

1,006931669 

1,004615794 
1,002305238 



1,002074475 
1,001843766 

1,001613109 

1,001382506 

1,001151956 

1,000921459 

1,00069101^ 

1 ,00046062*3 

1,000230285) 



,00009 
8 

7 
6 

5 
4 

3 

2 

I 



,000009 
8 

7 
6 






000207254 
OCO184224 
000161 194 
0001 38165 
0001 151 36 
000092106 
COO069080 
00004605 3 
000023026 



000020724 
0000 1 84 2 I 
OOOO16118 
OOOO13816 
OOOOII513 
0000092 10 
000006908 
000004605 
000002302 



Loji, 



,0000009 
8 

7 
6 

5 

4 

3 

2 

1 



oooooooq 
8 

7 
6 

5 
4 

3 

2 

I 



Nat. Numb. 



,000002072 
,000001842 
,000001611 
,000001381 
»oooooii5i 
,00000092 1 
,000000690 
,000000460 
,000000230 



,000000207 
,000000184 
,000000161 
,000006138 
,000000115 
,000000092 
,000000069 
,000000046 
,00000002 3 

where. 



L O 6 AR ITHMS« 115 

v^here, becaufe the logarithms in each clafs are the continual mul- 
tiples I, 2y 3, &c, of the loweft, it is evident that the natural num<» 
bers are fo many Icales of geometrical prpportionals, the loweft be- 
ing the common ratio, or the afcending numbers are the i, a, 3, 
&c powers of the loweft, as exprefled bv the figures i, 2, 3, &c of 
their correfponding logarithms. Alfo tne laft number in the firft, 
ftcond, third, &c clafs, is the loth, icx)th, loooth, &c root of 10 ; 
and any number in any clafs is the loth power of the correfponding 
number in the next following clafs* 

To find the logarithm of any number, as fuppofe of 2000, by 
this t^ble t Look in the firfl clafs for the number next lefs than the 
Arft figure 2, and it is 1,995262315, againft which i« 3 for the 
iirft figufe of the logarithm fought. Again, dividing 2 the number 
proposed by 1,995262315 th^ number found in the table, the quo* 
tient is 1,002374467 ; which being looked for in the fecond clafs 
of the table, and finding neither its equal nor a lefs, o is therefore 
to be taken for the fecond figure of the logarithm ; and the fame 
quotient 1)002374467 being looked for in the third clafs, the next 
lefs is thete found to be 1,002305238, againft which is I for the third 
figure of the logarithm ; and dividing the quotient 1,002374467 by the 
faid next lefs number 1,002305238, the new quotient is 1,000060070; 
which being fought in the fourth clafs gives o, but fought in the 
fifth clafs gives 2, which arc the fourth and fifth figures of the 
logarithm lought :>again, dividing the laft quotient by 1,000046053 
the next lefs number in the table, the quotient is 1,9900230159 
which gives 9 in the 6th clafs for the 6th figure of the logarithm 
ibught : and again dividing the laft quotient by 1,000020724 the 
next lefs number, th'; quotient is 1,000002291, the next lefs thaA 
which in the 7th clafs gives 9 for the 7th figure of the logarithm : 
and dividing the laft quotient by 1,000002072^ the quotient is 
1,000000219, which gives 9 in the 8th clafs for the 8th figure of 
the logarithm : and again the laft quotient 1,000000219 being di- 
vided by 1,000000207 the next lefs, the quotient 1,000000012 gives 
5 in the fame 8th dais, when one figure is cut ofF, for the 9th figure 
of the logarithm fought. All which figures colleSed together give 
3,301029995 for Briggs's logarithm of 2000, the index 3 being fup- 
piied ; which logarithm is true in the laft figure. 

To find the number anfwering to any given loga- 
rithm, as fuppofe to 3,3010300 : omitting the cha- 
rafteriftic, againft the other figures 3,0,1,0,3,0,0, as 
in the firft column in the margin, are the feveral 
numbers as in the 2d column, found from their re- 
{pcStivc I ft, 2d, 3d, &c claffes 5 the efFe^ive num- 
bers of which multiplied continually together, the 
laft produd is 2,000000019966, which, becaufe the charaderiftic 
b 3, gives 2000,000019966 or 2000 only for the required number 
BflMwering to the given logarithm. 

0,2 Of 



1,995262315 

o 

1,002305238 

o 

1 ,000069080 

o 



ii6 Construction ot ' 

« 

Of Mr. Jones's Afeth&d. 

In the 6ift volume of the Ph^Jofophical TranfaAions, is a fmzlt 
paper on logarithms, which had been drawn up and left unpub- 
Kfhed, by the learned and ingenious William Jones, £fq« The 
method contained in this memoir, depends on an application of the 
dodrine of fluxions, to fome properties drawn from the nature of 
the exponents of jfx>wers. Here all numbers are confidered as fonse 
certain powers of a conftant determinate root : fo any number m 
may be confidered as the z power of any root r, or that xzzr^ is a 
general expreffion of all numbers in terms of the conftant root r and 
a variable exponent x. Now the index % being the logarithm of 
the number Xy therefore to find this logarithm, is the fame thing as 
to find what power of the radical r is equal to the number x. 

From this principle, the relation between the fluxions of zny 
number x and its logarithm » is thus determined : put r =r i -}- » » ' 

then is x = r^ = i + «i *, and x + x = r+^» + » =r i + «V« y^ 

1 +«V» = * X I +n^''» = (by expanding r -h »\», omitting the ad, 

3d, &c powers of x, and writing y for ^x + xx x : y + f j* 

therefore x = axiy putting a for the feries ? + f f* + i f * &c^ 
or/Jr= xij putting/= — . 

Now when r = i + « = 10, as in the common logarithms of 
Briggs^s form; then »=9, ^=,9, and the feries f + f f * + f f * 
&c §ives a = 2,302585 &c, and therefore its reciprocal / = 
5434294 &c. But if a = 1 =}; the form will be tbat of Napier's lo- 
garithms. 

From the above form *i =/x. or i = ^, are then deduced man, 

curious and general properties of logarithms, with the feveral feries 
heretofore given by Gregory, Mercator, Wallis, Newton, and 
Halley. But of all thefe feries, that one which our author feleds 

for conftruding the logarithms, is this ; putting iV' => ^ ^ ^ the 
logarithm of -J is = 2/ x : N+ iN^+fN^+^^lfl &c in the 
cafe in which r — ^ is = i, and confequently then AT = ^ or 
-— j— ; which feries will then converge very faft. 

Hence, having ^iven any numbers, ^, ^, r, &c, and as many 
ratios tf, *,vf, &c, compoled of them, the difference between the 
two terms of each ratio being 1 j as alfo the logarithms J^ £, C, &c 

of 



LOGARITHMS. txy 

of tbofe ntioe given : to find the logarithms P, ^, H^ &cof tliofe 
numbers ; luppofing/ = i. For inftance> if ^ = 29^ = 3, r=5; 

rithms ^, £j Cy of thefe ratios a, 3, ^, being found by the above 
feries, from the nature of powers we have thefe three equations, 
Jsz 2^— 3P pwhichcqua-}P=3^+45+2C=log. of 2, 

B=4P— ^— ^> tionsre- {•^=5y/+6P+3C=log, of 3, 
C= iR — £- 3Pi duced give 3 ^=7^+95+5^=105. of 5. 

And hence P + ^ = 10^ + 13P + yCis = the logarithm of 2 X 5 
or zo. 

An elegant tra£t on logarithms, as a comment on Dr. Halley's 
method, was alfo given by Mr. Jones in his Synods Falmarhrum 
Mdthefeosy publiihed in the year 1706* And in the Philofophical 
Tranfadtions he communicated various improvements in gonio* 
metrical properties, and the feries relating to the circle and to tri- 
gonometry. 

The memoir above defcribed was delivered to the Royal Society 
by their then librarian, Mr. John Robertfon, a worthy, ingenious^ 
and induftrious man ; who alfo communicated to the Society feveral 
little traAs of his own relating to logarithmical fubje£(s : he was 
alfo the author of an excellent Treatife on the Elements of Naviga- 
tion in two volumes ; and he was fucceflively mathematical mafter 
to Chrift's hofpital in London ; head mafter to the royal naval aca- 
demy at Portfmouth ; and librarian, cleric^ and houfekeeper to the 
Royal Society ; at whofe houfe, in Crane Court, Fleet otreet, he 
died in 1776, aged 64 years. 

And among the papers of Mr. Robertfon, I have, fince hit 
death, found one containing the following particulars relating to 
* Mr. Jones, which I here infert, as I know of no other account of 
his life, &c. and as any true anecdotes of fuch extraordinary men muft 
always be acceptable to the learned. This paper is not in Mr. Ro- 
bertion's hand writing, but in a kind of tunning law*hand, and is 
iigned R. M. 12 Sept. 1771. 

*.' William Jones, Efq. F. R. S. was born at the foot of Boda- 
von mountain, [Mynydd Bodafon] in the parifh of Llanfihajlgel 
tre'r Bardd, in the iflc of Anglefey, North Wales, in the year 1675* 
His father John George * was a farmer, of a good family, . beine 
defcended from Hwfa ap Cynddelw, one of the 15 tribes of Nortn 
Wales. He gave his two ions the common fchool education of the 
country, reading, writing, and accounts, in Englifh, and the Latin 



* '< It is the caftom in feveral parts of Wales for the name of the father to becoin« 
the famame of his children. John George the father was commonly called Sion Siors of 
liaobabo, to which pariih he moved, and where bis children were brousht up." 

grammar 



itf CONSTRUCTION* OP 

|:rammar. Harry his fecond fon took to the fanning bufinefs ; but 
William the eldeft, having an extraordinary turn for mathematical 
ftudieSy determined to try his fortune abroaa from a place where the 
fame was but of little fervice to him ; he accordingly came to Lon- 
don, accompanied by a young man Rowland Williams, afterwards 
an eminent perfumer in Wych Street. The report in the country 
is, that Mr. Jones foon got into a merchant's counting houfe, .and 
fo gained the efteem of his mafter, that he gave him the command 
of a Ihip for a Weft India voyage ; and that upon his return he fee 
up a mathematical ichool, and publifhed his book of navigation *; 
and that upon the death of the merchant he married his widow : that 
Lord Macclesfield's fon being his pupil, he was made fecretary to 
the chancellor, and one of the D, tellers of the exchequer — and 
they have a ftory of an Italian wedding which caufed great diftuf- 
bance in Lord Macclesfield's family, but compromifed by Mr« 
ijones; which gave rife to a faying, that Macclesfield was the 
making of Jones, and Jones the making of Macclesfield," 

Mr. Jones died July 3, 1749, being vice-prefident of the Royal 
Society; and left one (daughter, and his widow with child, which 
proveo a fon, who is the prefent Sir William Jones, now one of the 
judges in India, and highly efteemed for his great abilities, exten- 
five learning, and eminent patriotilm. 

Of Mr* Andrew Reid and Other si 

Andrew Reid, £fq. publifh^ in 1767 a quarto traft. under the 
title oi An EJfay on Logarithms^ in which he alfo fhews the compu« 
tation of logarithms from principles depending on the binomial 
theorem and the nature of the exponents of powers, the logarithms 
of numbers being here confidered as the exponents of the powers of 
10. He hence brings out the ufual ieries for logarithms, and largely 
exemplifies Dr. Halley's moft fimple conftruAion. 

Befides the authors whofe methods have been here particularly de- 
fcribed, many others have treated on the fubjeAs of logarithms, and 
of the fines, tangents, fecants, &c; among the principal of whom 
are Leibnitz, Euler, Maclaurin, Wolfius, and profefibr Simfon in 
an elegant geometrical traifton logarithms, contained in his pofthu- 
mous works, elegantly printed in 4to. at Glafgow in the year 1776, 
at the expenceof the very learned Earl Stanhope, and by his Lord£bip 
difpofed of in prefents among gentlemen raoft eminent for mathema* 
tical learning. 



* This trad on navigation^ intituled, '< A New Compendium of the whole Art of 
practical Navigation," was publilhed in 1702, and dedicated ^ to the reverend and 
learned Mr. John Harris M. A. and F. R. S." the author I apprehend of the " U iver* 
pX Diaionary of Aits and Sciences/' under whofe roof Mr. Jones fays be compofed tbo 
^'^ tfcataiip on Navigauoo* 

Of 



LOGARITHMS. li$ 

Of Mr, DodforCs Anti'logarithmic Can$n. 

The only remaining confiderable work of this kind publlflied, that 
I know of, is the Anti-logarithmic Canon of Mr. James Dodfon, s^ 
very ingenious mathematician, which work he publifhed In folio in 
the year 1742; a very great performance, containing all logarithms 
under ioocx)0, and their correfponding natural numbers ton places 
of figures, with all their differences and the proportional parts ; the 
whole arranged in the order contrary to that ufed in the common 
tables of numbers and logarithms, the exad logarithms being here 
placed firfl, and increafii^g contincially by i, from i to icx)ooo, and 
their correfponding neareft numbers in the columns oppofite to 
them \ and by means of the differences, and proportional parts, the 
logarithm to any number, or the number to any logarithm, each to 
ji places of figures, is readily found. This work contains alfo, 
befides the conftruAion of the natural numbers to the given loga- 
rithms, ** precepts and examples, (hewing feme of the ufes of loga** 
rithms, in facilitating the mof( difficult operations in common arith* 
metic, cafes of interef)r, annuities, -menfuration, &c ; to' which is 
prefixed an introdudion, containing a fhort account of logarithms^ 
and of the moft confiderable improvements made, iince their inven- 
tion, in the manner of conftru^ing them.'' 

The manner in -which thefe numbers were conftrudled, confifls 
chiefly in imitations of fome of the methods before defcribed by 
Briggs, and is nothing more than generating a fcale of iooocx> 
geometrical proportionals from 1 the leaft term to 10 the greatefl, 
each continued to 11 places of figures; and the means of effecting 
this are fuch as eafilv flow from the nature of a feriesof proportionals, 
and are briefly as follows. Firft between x and ip are interpofed 9 
mean proportionals; then between each of thefe ji terms there are 
interpofed 9 other means, making in all 101 t^rms ; then between 
each of thefe a 3dfet of 9 means, making in all lopi terms; again 
between each of thefe a 4th fet of 9 means, makitig in all loooi 
terms ; and laftly between each two of thefe terms, a 5th fet of 9 
mean$, making m alliooooi terms, including both the i and the 
JO. The firfl four of thefe 5 fets of means, are found each by one 
extradlion of the loth root of the greater of the two given terms, 
which root is the leafl mean, and then multiplying it continually by 
itfelf according to the number of terms in the fedion or fet; and 
the 5th or lafl feSion is made by interpofing each of the q means by 
help of the method of differences before taught. Namely, putting 

10 the greatefl term =,A, A" = B, B"=C,C'^'^ = D, D"^* 

I 
r= E, and E"^ =s F; now extraSing the loth root of A or 10, it 

gives I125892541 18 = B = A^^ for the |eafl of the ifl fet of means; 
and then multiplying it continually by itfelf, we have B, B%^', B% 

&c 



lao 



CONSTRUCTION OF 



&c to B'«> = A for all the lo terms: 2dly, the loth root of 

1,2589254118 gives 1,0232929923 - C = B^^ = A'^^forthcleaft 
of the 2d clafs of means, which being continually multiplied gives 
C, CS C\ &c to C»«» = B>o r= A for all the 2d clafs of 100 terms: 
3dly, the loth root of 1,0232929923 gives 1,0023052381 = D =3 
C tV _ griw _ ^rzhz f^^ ^^^ j^^jj ^f ^y^ ^j ^j^f^ ^f ^eans, which 

being continually multiplied gives D, D», D3, &c to D»<^ = C»«> 
= B*^ s= A for the 3d clafs of 1000 terms : 4thly, the loth root 

1,0023052381 gives 1,0002302850 = E = D^ = c"^®^ =s B""^^** 
^j^TZTSJsv £^y ^jjg jg^j^ Qf (|jg ^^h clafs of means, which being con- 
tinually multiplied gives E, ES E', &c to E'^^o - Diooo =- C«<» 
=: B<^ :? A for the 4th clafs of 10000 terms. Now thefe 4 clafles 
of terms thus produced, require no lefs than 1 1 1 10 multiplications of 
the leaft means by themfelves ; which however are much facilitated 
by making a fmall table of the firft jo or even 100 produSs of the 
conftant niultiplier, and from thence only taking out the proper 
lines and adding them together: and thefe 4 clafles of numbers al« 
ways prove themfelves at every loth term, which muft always ^grce 
with the correfponding fucceiSve terms of the preceding clafs. The 
remaining 5th clafs is conftru6^ed by means of differences, being 
niuch eafier than the method of continual multiplication, the ift and 
2d differences only being ufed, as the 3d difference is too fmall to 
enter the computation of the fets of 9 means between each two 
terms of the 4th clafs. And the feveral 2d differences for each of thefe 
fets of 9 means, are found from the properties of a fet of proportion- 
als i,r, r», r3,^&c, as 
difpofed in t}ie ift co- 
lumn of the annexed 
table, and their feve- 
ral ordersof differences 
as in the other columns 
of the table ; where it 
is evident that each co- 
column, both that of the given terms of the progreffion, and thofe 
of their orders of differences, forms a fcale of proportionals, having 
the fame common ratio r ; and that each horizontal line or row forms 
a geometrical progreffion having all the fame' common ra tio r — I, 

which is alfo the jfl difference of each fet of means ; fo r — i^* is 
the ift of the 2d differences, and which is conftantly the fame, as 
the 3d differences become too fmall in the required terms pf our pro- 
greffion to be regard ed, at leaft near the beginning of the table : hence, 
like as i, r — i, an d r*— i ^* are the ift te rm with its ift and ad diffe- 
rences; fo r", r", r — X, and r». r — i^* are any other term with its 
ift and 2d differences* And by this rule the lit and ad differences 
sure to be fpvnd fpr every fet of 9 meansy viz. multiplying the ift 

tern^ 



Terms 1 iddif. i addif. | 3d dif. | &c 


I X 1 r— IX 1 r— P* X {r— i)' X | 


I 
r 
r» 
r? 
&c 


I 
r 

r» 
&c 


I 

r 

r» 

r» 

&c 


I 

r 

&c 






LOGARITHMS. xzx 

term of any clafs (which will be the feveral terms of the feries E, E% 
E', &c, or every loth term of the feries F, F% F', &c) by r — * i or 
F — I for the ift dilFerence, and this multiplied by F-— I again for 
the true 2d difference at the beginning of that clafs. Thus the 
loth root of i,cx)0230285o or £ gives 1,000023026116 for F or 
the ift mean of the loweft clafs, therefore F— i =:rr— 1 = 
,000023026116 is its ift diiFerencCy and the fquare of it is r — 0* 
=,0000000005302 its 2d difference; then is ,0000230261 i6F'o^ 
or ,0000230261 1 6E" the ift difference, and ,0000000005 302F«>" 
or ,ooooooooo5302E^'> ^ is the ad difference at the beginning of the 
nth clafs of decades. And this 2d difference is ufed as tjie conftant 
2d dHFerence. through all the 10 terms, except towards the^nd of the 
table where th^ differences increafe faft enough to require a fmall 
corredion of the 2d difference, and which Mr, Dodibn effeds by 
talcing a mean 2d difference among all the 2d differen ces in this 
manner ; havin g fou nd the feries of ift differences F — i . E% 
F — I. £« + ', F — I • E«+*, &c, take the differences of thefc, 
and yV of them will be the mean 2d diffcren<;es to "be ufed, namely 

F "^ r -_— — F — I — — — ^— — — — 

. K« + 1 — j^if, • E" + » — t" + '• &c are the mean 

10 ' 10 ' 

2d differences. And this is not oply the more exaft but alfo the eafier 

way. The common 2d difference and the fucceffive ift differences 

are then continually added through the whole decade, to give the 

fucceffive terms of the required progreffion. 



JDESCRIPTIO^I 



( «3t ) 



P ES CR I PTION 



AND USE OF 



LOGARITHMIC TABLES. 



ALTHOUG H the nature and conftrudion of logarithms have been 
pretty fully treated in the preceding hiftory of fuch numbers, 
where the more learned and curious reader will find abundant fatis- 
faAion, I fhall here give a brief, eafy, and familiar idea of thefe 
matters, for the prad^ical ufe of young iludents in this fubjeiS:. 

The Definition and Notation of Logarithms. 

Logarith^msare the indices, or arithmetical feries of numbers, adapt* 
cd to the terms of a geometrical feries, in fuch fort that o correfponds 
to, or is the index of, i in the geometricals. 

.pi Co 12 ? 4 S» &c. indices or logaritbros. 



24 8 i6 32, &c. geometric progreliion, 

234 5, &c. indices or logaritl 

9 27 81 243, &c. geometric feries« 



I o I 2 3 4 5, &c. indices or logarithms, 

or \^ 123 4 5» &c. indices or logarithms, 

£ 1, 10, 100, 1000, 1 0000, 1 00000, &c. geometric feries. 

Where the fame indices ferve equally for any geometric feries; and 
from which it is evident that there may be an endlefs variety of 
fy ft ems of logarithms to the fame common numbers, by varying the 
2d term 2, or 3, or 10, &c, of the geometric feries; as this will 
change the original feries of terms whofe indices are the whole num- 
bers, I, 2, 3t &c; and by interpolation the whole fyftem of num- 
bers may be made to enter the geometrical feries, and receive their 
proportional logarithms, whether integers or decimals. 

Or, the logarithm of any number is the index of that power of 
ffome other number, which is equal to the given number. So if N 
be =: r», then the logarithm of N is «, which may be either pofttive 
or negative, and t any number whatever, according to the different 
fyftems of logarithms. When N is i, then « = o, whatever the va- 
lue of r is ; and confequently the logarithm of i is always in every 
fyftem of logarithms. When n is = i, then N isrrr; confe- 
quently r is always the number whofe logarithm is i in every fyf- 
tem. When r is = 2*7 1828^828459 &c, the indices are the hyper- 
bolic logarithms, fuc h as in ou r 7th table; fo that n is the hyper- 
bolic logarithm of 2718 &c«^^ But in the common logarithms r 

ii5 



T H E T A B L E S* 115 

is =r 10 ; fo that the common logarithm of any number (lO") is 
(n) the index of that power of 10 which is equal to the faid num« 
ber. So 1000, being the ^ power of 10, has 3 for its logarithm; 
and if 50 be =: 10^*^9897, then is 1*69897 the common logarithm of 
50. And hence it follows that this decimal feries of terms 

10* y 10', 10% 10% 10% 10 ~% 10-*, 10-', 10-4 , 

or icooo, 1000, 100 > 10 , I , •!, 'oi ,'ooi, •oooi , 
have 4 , 3,2,1,0,-1 f -2 , -3, -4 , 

rcfpeftively for their logarithms. 

The logarithm of a number comprehended between any two 
terms of the firft ieries, is included between the two correfpond- 
ing terms of the latter, and therefore that logarithm will confift 
of the fame index (whether pofitive or negative) as the lefs of thofe 
two terms, together with a decimal fradlion, which will always be 
pofitive. So the number 50, falling between 100 and 10, its loga- 
rithm will fall between 2 and I, and is =: 1*69897, the index of the 
lefs term together with the decimal '69897 : alfo the number '05, 
falling between the terms *i and 'Of, its logarithm will fall between 

— I and — 2, and is indeed z: — 2 + '69897, the index of the lefs 
term together with the decimal '69897. The index is alfo called 
the charaSeriftic of the logarithms, and is always an integer, ei- 
ther pofitive or negative, orelfe = o; and it (hews what place is 
occupied by the firft fignificant figure of the given number, either 
above or below the place of units, ^being in the former cafe + or po- 
fitive, in the latter — or negative. 

When the charafteriftic of a logarithm is negative, the fign — is 
commonly fet over it, todiftinguifh it from the decimal part, which 
being the logarithm found in the tables, is always pofitive: fo 

— 2 + '69897, or the logarithm of -05, is written thusT*69897- But 
on fome occaiions it is convenient to reduce the whole expre&on to a 
negative form ; which is done by making the charafteriftic figure 
lefs by I, and talcing the arithmetical complement of the decimal, 
that is, beginning at the left hand, fubtradt each figure from 9, ex- 
cept the laft fignificant figure, which fubtraS from 10 5 fo (hall the 
remainders form the lop:arithm iniirely negative/ Thus the loga- 
rithm of '05, which is'2'69897 or — 2 + '69897, is alfo expreflcd 
by — i«30iO3, which is wholly negative. It is alfo fometimes 
thought more convenient to exprefs fuch logarithms wholely as po- 
fitive, namely by only joining to the tabular decimal the comple- 
ment of the index to lO; in which way the above logarithm is ex- 
preiTed by 8*69897; which is only increafing the indices in the 
fcale by 10. It is alfo convenient, in many operations with loga- 
rithms, to take their arithmetical complements, which is done by 
beginning at the left hand, and fubtrafling every figure from 9, but 
the laft figure from 10: fo the arithmetical complement 
ofi'69897 f and of 2-69897 7 where the index — 2, being nega- 
is 8'30i03, \ it is ii '30103, J tivc, is added to 9, and makes ii, 

^ R 2 • nc 



124 PROPERTIES &c OF 

Thi Properties of Logarithms. 

From the definition of logarithms, either as being the indices of 
a feties of geometricals, or as the indices of the powers of the fame 
root, it follows that the multiplication of the numbers will anfwer 
to the addition of tl^eir logarithms ; the divifion of numbers to the 
fubtra£lion of their logarithms ; the raifing of powers, to the multi* 
plying the logarithm of the root by the index of the power ; and 
the extra£ling of roots, to the dividing the logarithm of the given 
number by the index of the root required to be extraded: So 
ift. h. ah ox axhis^h. a-^-h. b 

L. i8 or 3x6is =1 L. 3+ L. 6 

L. 5 X 9 X 73 is = L. 5 + L. 9 + L. 73 
2d.L. a -^ b is =rL. a— -L.^ 

L. 18 4. 6 is zrL. 18 — L. 6 

L. 79x5-1. 9 is = L. 79 4 L. 5— L. 9 

L. ior i-^ 2 is = L. I — L. 2=0 — L. 2 = — L, 2 

L. — or I ^ « is = — L. » 
n 

3d.L.r«i8 = «L. rjL.r '»orL. V ris= — L.r; L.r « isz: — L.r. 

fi u 

L. 2*is=:6L.2;L. 2'orL. v/iis = f L. 2; L. 2 is n 3 L.2. 

So that 4ny number and its reciprocal have the fame logarithm, 
but with contrary figns ; and the fum of the logarithms of any 
' number and its complement, is equal to o. 

To conJlru£i Logarithms. 

It has been fliewn in the foregoing hiftorlcal part, that the loga- 
rithm of- is = ^ X : 1+ -A. + * + JL &c; where % is the 

fum, and x the difference of a and h \ alfo m = 2*302585092994 
&c, the hyp. logarithm of 10. Therefore if a and b be any two 
numbers differing only by unity, io that x ox b^^a may be = i | 

then fliall the logarithm ofibe = L. «+— x :-4.~+ — &c. 

Which gives this rule in words at length : call « the fum of any 
number (whofe logarithm is fought) and the number next lefs by 
unity; divide '8685889638 &c (or 2~ 2.3025 &c) by z, and re- 
fcrve the quotient j divide the referved quotient by the fquare of s, 
and referve this quotient ; divide this laft quotient alfo by the fquare 
of z, and again referve this quotient : and thus proceed continually 
dividing the lafl quotient by the fquare of z> as long as diviiion can 

be 



L06ARI THMS. 



"$ 



be made. Then write thefe quotients orderly under one another, 
the firft uppermoft, and divide them rcfpeftively by the uneven 
numbers I, 3, j, 7) 9, 1I9 kc, as long as divifion can be made; 
that is, divide the firft reserved quotient by i, the 2d by 3, the 3d 
by 5, the 4th by 7, &c. Add all thefe laft quotients together, and 
the fum will be the logarithm of ^-f ^ ; and therefore to this loga- 
rithm add alfo the logarithm of a the next lefs number, and the fum 
will be the required logarithm of b the number propofed* 



£x* I . To find the Log. of 2 • 
Here the next lefs number is i, & 3 4* ^ 
== 3 =rzy whofe fquare is 9. Then 



3)-868588964 
9)"i895296c4 
9) 32169962 

9) 3574440] 7) 
397160 
44129 
4903 



9) 
9) 
9) 
9) 
9) 



01 



i)-289S296s4(-289S29654 
3) 32i69962( 10723321 

5) 3574440{ 714888 

397i6o( 56737 

44i29( 4903 

4903( 446 

S4S( 42 
6i( 4 
Log. J - - •30102999s 
Add L. I - -000000000 

Log. of X- '301029995 



Ex. 2. To find the Log* of i* 
Here the next lefs number is 2, and 2 -)* 5 
= 5 = 2, whofe fquare is 25, to divide 
by which always multiply by '04. Thea 



9) 

XI) 

'3) 
'S) 



5 )-868588964 
25)-i737i7793 
25) 6948712 

25) ^77948 
25) 11118 

25) 44$ 
18 



i)-i737i7793(-i737i7795 
3) 69487 I 2( 2316237 

S) 277948( SS$9» 
7) iiii8( 158S 

9) 448{ 50 

11) i8 ( z 

L.4 - * '176091269 
L« 2 add - •301029925 

L. 3 - . '477»2X2S$ 



Then becaufe the fum of the logarithms of numbers gives the 1o« 
garithm of their produd, and the difference of the logarithms gives 
the logarithm of the quotient of the numbers » from the above two 
logarithms, and the logarithm of 10, which is i, we may raife a 
great many logarithms thus : 



Ex. 3. Becauie 2x2: 
toL. 2 • - 
add L. 2 

fum is L. 4 

Ex. 4. Becaufe 2x3 
to L. 2 
add L. 3 

fura is L. 6 

Ex, 5. Becaufe 2^ = 
L. 2. 
mult, by 

gives L. 8 - 



4, therefore 

'301029995I 

•301029995I 



»6020 



5999^7 



= 6, therefore 

•301029995 

778151250 



Ex. 6. Becaufe 3^ 
L.3 - 
mult, by 

gives L. 9 



9, therefore 
•477121254/ff 

2 

'954242509 



8, therefore 
•3010299951 

L 

•903089987 



Ex. 7. Becaufe '/ =: c, therefore 
from L. 10 «• 
take L. 2 

leaves L. 5 



5> 
1*000000000 

-301029995* 
•698970004^ 



Ex. 8. Becaufe 1 2 
toL. 3 
add L. 4 

gives L. 12 



3x4, therefore 

•477121255 
•6020^9991 

i*079i8i246 



And thus by computing, by the general rule, the logarithms of 
the other prime numbers 7, 11, 13, 17, 19, ^3, &c ; and then ufing 
compofition and divi/ion, we may eafily find as many logarithms as 



we pleafe, or may fpeedily examine any logarithm in the table. 



THE 



"6 . DESCRIPTION AND USE 



THE DESCRIPTION AND USE OF THE 

TABLES. 

np H E following colIcAion con/ifts of various tables, in this 
•* order, viz. i, A large table of logarithms to 7 places of figures; 
2, A table for finding logarithms and numbers to 20 places ; 3, Loga- 
rithqis to 20 places, with their ift, 2d, and 3d differences; 4^ An- 
other table of logarithms to 20 places^ with their ifl, 2d, and 34 
differences ; 5, Logarithms to 61 places; 6, Another table of loga- 
rithms to 61 places, with their ifl, 2d, 3d, and 4th differences ; 7^ 
Hyperbolic logarithms ; 8, Logiftic logarithms ; 9, Logarithmic fines 
and tangents to every fecond of the' firfl 2 degrees ; 10, Natural and 
lo^rithmic fines, tangents, fecants, and verfed fines, with their 
differences to every minute of the quadrant. After which fol- 
low feveral fmaller tables ; as a table of the lengths of circuhr arcs ; 
a traverfe tabl^, or table of difference of latitude and departure, to 
every degree and quadrant point of the compafs ; a table for chang- 
ing the common logarithms into hyperbolic logarithms ; and a table 
of the names and number of degrees &c. in every point of the com- 
pafs ; as alfo lifts of errata in various works of this fort. Of eaci^ 
of which in their order. 

Of the large Table of Logarithms* 

The firfl is the large table of logarithms to all numbers from i to 
lOOOOO, by which may be found the logarithm to any number and 
the number to any logarithm to 7 places of figures. This table 
confifls of two parts; the firfl contains, in 4 pages, the firfl 1000 
numbers with their correfponding logarithms iii adjacent columns ; 
the fecond contains all the 1 00000 numbers aqd their logarithms, 
with the differences and proportional parts, difpofed as follows : in 
the lA column of each page are the firfl 4 figures of the numbe]:s, 
and along the top and bottom of the columns is the 5th figure, in 
which columns are placed all the logarithms, the firfl 3 figures of 
each logarithm being at the beginning of the lines in the firfl column 
of logarithms, figned o at the top and bottom, and the ' other 4 
figures in the remaining columns. After the 10 columns of loga- 
rithms flands their column of differences, figned D ; and laflly after 
that, as alfo at the bottom of lome pages, the column of propor- 
tional parts, figned pro. pts. fhewing what proportional part of 
each difference correfponds to i, 2, 3, &c, the whole difference an- 
fwering to 10 ; or fhewing the t^, ^%^ t\, &c, of the differences. 

Note^ the logarithms in thefe columns are all fuppofed to be deci- 
mals, and their correfponding natural numbers may be either inte- 
gers or decimals or mixt numbers^ for the fame figures, whatever be 

, their 



OFTHETABLES. 127 

their denomination, have the fame decimal logarithm, and tkefe 
differ only in the index or charaSeriftic, which is the integer num- 
ber to be prefixed to the decimal pTart of the logarithm ; and this is 
always the number which exprefTes the diftance of the higheft deno- 
mination, or left-hand figure, of the natural number, from the units 
place. So that if the natural number conAft of only one place of in- 
tegers, the index of its log, will be O; if of 2, 3, 4, 5, &c, the 
index of its logarithm will be refpe<^ively i, 2, 3, 4, &c, being 
I lefs than the number of integer places : and the fame figures made 
negative will give Che index of the logarithm of a decimal, viz. if 
the natural number be a decimal, and its firft fignificant fi^re be 
in the place of primes, 2ds, ^ds, 4.ths, &c. the index of its logarithm 
will be refpeftively 1,2,3,4, &c. till the figure which ex- 
prefles the diftance of the firit place of the natural number from 
the units place, but with a negative fign, as the number is below 
the place of units, the fign being written above the index inflead of 
before it, as that part only of the logarithms is to be confidered as 
negative, the decimal part of it being always affirmative. And in 
the arithmetical operations of addition and fubflradlion with loga- 
rithms, the negative indexes will have the contrary effed to that of 
the decimal part of the logarithm, viz. when the logarithm is to be 
added, the figure of the negative index muil be fubftraded, bf vice 
vtrfa. Hence if 423409 be the tabular or 
decimal part of the logarithm belonging to 
the figures 2651, without any regard to their 
particular denominations ; then according as 
they are varied with refpeft to the number 
of decimals, as in the ift annexed column, 
the index of their logarithm, and the com- 
pleat logarithm,will vary as in the 2d column 
here annexed. And hence, like as when the 



Number 
26 c I 
2651 
2651 



Logar. 

2-4234097 
J '42 34097 



2651 0*4234097 
•2651 
•02651 
•002651 



i;42 34097 
2^4234097 
3-4234097 



natural number is given, we find the index of its logarithm by 
counting how far its firft figure on the left hand is from the units 
place ; fo when a logarithm is given, the denominations of the 
figures in its natural number will be found by placing the decimal 
point fo, that the number of integer places may be I more than 
that of the index when pofitive, or by fetting the firft fignificant 
Jigure in that decimal place, which is exprelTed by the number of 
the index when negative. 

Of finding the Logarithm of a given Number y or the Number to a given 

Logarithm. 

I. To find the Logarithm of a Number confijiing of i Figures. 

Find the number in the column of numbers in one of the firft 4 
pagiss of the table, and immediately on the right of it is its logarithm 
fought. So the logarithm of 72 is i-85733Z5^ and the logarithm 
of 3'33 is 05224442, when the proper index is fupplied. 

2. To 



12S DESCRIPTION AND US£ 



2U To find the Logarithm of a Number confijling of 4 Places^ 

In the firft column (fignedN) in fomeoneof the pages of the 
table after the iirft four, find the given number, then againft it in 
the 2d column (figned o) is the logarithm fought. So the logarithm 
of 22S4is 3'3529539> ^^^ ^^^^ ^^ 31^32 is 1*4958218. 

3. To find the Logarithm of a Number confi/iing of ^ Places. 

Find the firft 4 figures of the given nurhber in the firft column as 
Before, and the 5th figure at the top or bottom ; then the 7 figures 
of the logarithm are found in two columns on the line of the firft 4. 
figures of the given number> viz. the firft 3 figures of the logarithm 
are the firft 3 common figures of the 2d column (figned o), and the 
laft 4 figures are on the fame line, but in the column figned with 
the 5th figure of the given number. So the logarithm of 23204 is 
4/3655629, and that of 746-40 is z'8V297i6, and that of -083178 is 

2-9200085. 

NotOy When the laft four figures of the logarithm begins with a 
cipher, or any figure lefs than the laft four in the 2d column begins 
with, then the firft 3 common figures are thofe in the next lower 
line : fo in the laft example the firft 3 common figures are 920, and 
not 919* 

4* To find the Logarithm of a Number of 6 Places^ 

Find the logarithm of the firft 5 figures by tbe laft article, and 
talce the difference between that logarithm and the next following 
logarithm, or (which is the fame) find the difference neareft oppo* 
£te in the laft column but one, figned D ; then under that difference 
in the laft column (of proportional parts) and againft the 6th figure 
of the given number, is the part to be added to the logarithm before 
found for the firft 5 figures, the fum being the logarithm fought. 
So to find the logarithm of 3409*26. The logarithm of 34092, the 
firft 5 figures, being 5326525, and the common difference 127, un^ 
der which and againft 6 in the laft column is 76, ,which being added 
to the former logarithm, and the proper index prefixed, we have 
3*5326601 for the whole logarithm required; 

5. To find the Logarithm of a Number of^ Placesm 

Find the logarithm of the firft five figures by the 3d article, and 
of the fixth figure by the 4th article ; then for the logarithm of the 
7th figure, divide its proportional part by 10, that is fet it one place 
farther to the right hand than the laft figure of the logarithm 
reaches ; add all the three together, and their fum will be the loga- 
xiihm required. 

Thus 



OF THE TABLES. 



129 



Thus to find the logarithm of 3-409264. ^^^^^ j^ 

The feveral parts being taken out according ^^^^^ K'^itcie 

to rule, and placed as in the margin, the 6 - - - ^ 76 

funt gives the whole logarithm fought. 4 . , j , 

Note, In the fame way we might take out 3.^09264- 0.C326606 
the proportional part of an 8th figure, divi- 
ding its tabular part by joo, or fetting it two places farther to the 
right hand than the firft logarithm. Or the whole proportional 
part for any number of figures above five, may be found at once by 
multiplying the common tabular difference of the loga- 
rithms, found as before, by all the figures after the 5th, 
cutting ofF from the produft as many figures as we mul- 
tiply by, and adding the reft to the logarithm of the firft 
5 figures before found. So in the laft example above, 
having found the common diflFerence 127, multiplying 
it by 64 the laft two figures, cutting ofF two, add the 
reft to the logarithm of the firft 5, as in the margin. 

For another example, fuppofe we wanted the logarithm of the 
following 8 figures 34.092648. The operation by both methods will 
be as below. 



127 
J64. 

508 
762 

S3^6;25 
0*5326606 



34092 



6 - r 

^ 

8 - 



5326525 
76 



5»i 
1,02 



34092648 



- 7*5326607 



127 
648 

1016 
508 
762 

82,296 
532652^ 



7 '53^6^07 the fame as the other. 



6. T0 find the Logarithm of a Vulgar FraSiiorty or of a Mixt Number, 

Either reduce the vulgar fraftion to a decimal, and find its lo- 
garithm as above. Or elfe (having reduced the mixt number to an 
improper fraftion), fubtraft the logarithm of the denominator from 
the logarithm of the numerator, and the remainder will be the lo- 
garithm of the fraftion fought 



Ex. u Tofindthclog.of-.'^oro-i875. 
From log. of 3 - o- 477>*»3 
Take log, of 16 - i. 204^^00 

Rem. log.ofy'scrMSys 7-2730013 



Ex. 2. To find the log. of 1 3I or ^^ 
From Ig. of 55 - - 1-7403627 
TaVe log. of 4 - - 0*6020600 
Ltavcslog.of^^^ori3'75 i'i383027 



7. To find the natural Number anfwering to any given Lcgarithm. 

Find the firft 3 figures, next after the index of the given loga- 
rithm, in the fecond column, figned o, and the other 4 figures on 
the fame line in one of the nine following columns ; if the figures 
of the logarithm be thus found exaSly, on the fame line in the firft 

S column 



130 DESCRIPTION AND USE 

column are the firft four figures of the natural number, and the 5th 
is at the top or bottom of that column in which the laft four figures 
of the log. were found. So to find the number anfwering to the 
logarithm 2-5890108. In pa. 64 I find the firft three figures 589, 
and in column 6 of the line above are found the other four 'OioS, 
(becaufe the firft three common figures are fuppofed to begin at that 
part of the line above where they are placed) ; then on the fame 
line in the column of numbers ftand the firft four figures 388-1, and 
6 at the top of the column, making in all 388* 16 for the number 
fought; having placed the decimal point fo as to make three inte- 
gers, being I moie than 2 the index of the given logarithm. 

But if the given logarithm be not found exaftly in the table, fub- 
traft the next lefs tabular logarithm from it, and look for the re- 
mainder in the proportional parts under the difference between the 
two tabular logarithms next lefs and greater than the given loga- 
rithm, and againft it, or the part next lefs, is a 6th figure tq be 
annexed to the five figures before found. And if the remainder be 
not found exa£Hy in the proportional parts, fubtraft the next lefs 
part from it, and annex a cypher to this 2d remainder, then againft 
the neareft proportional part (either greater or lefs) is a 7th figure 
to be annexed to the fix before found. And that figure will be the 
neareft to the truth in that place, either too much or too little. 

Ex. To find the number anfwering to the logarithm i'233567S. 
The next lefs tab. log. is the log. of 1 7 1 22 viz. ^3355^5 

I ft reip. 133 

r^. ,.«• C - - S for the part 127 

Tne difference is 254. \ ^d rem. 65" 

andthetableofpro.pts.gives^ . . 2 for the part - - 51 
So that the number fought is 17*12252, making two integers for the 

index I. 

Or the 6th and 7th figures may be found withciut the table of pro- 
portional parts, by dividing the firft remainder by the tabular diffe- 
rence, annexing one cipher to the dividend for each 254)1?^ CX)(ca 
figure to be found. So in the laft example, the re- \^\ 

mainder 133, with two ciphers annexed, being di- ^ — 

vided by the tabular difference 254, as in the margin, ^ 

the quotient gives 52 for the 6th and 7th figures, 5^^ 

the fame as before. 

In like manner may be found the numbers to the following loga- 
rithms. 

LogaTith.i'2345678j3*7343003jr-o92i4o6 2*3710468 4*6123004 3-2946809 
Numb. I7*i6200 ls*4237S8 ri236348 '^ 



•02355303 +^9S4"39 



1970-974 



OF 



OF THE TABLES. 



131 



OF LOGARITHMICAL ARITHMETIC. 

I. Multiplication by Logarithms. 

Add together the. logarithms of all the faftors, and the fum is a 
logarithm, the natural number correfponding to which will be the 
produ6): required. 

Obferving to add, to the fum of the afRrmative indices, what is 
carried from the fumiof the decimal parts of the logarithms. 

And that the difference betyveen the affirmative and negative in- 
dices Js to be taken for the index to the logarithm of the produft. 



Ex. I. To multiply a 3- 14 by 5'o62, 
23*14 its log.. is 1*3643634 
5-062 its log. is 0'704322i 

Product 11 7' 1 347 - - 2'o686855 

jB;r. 3. To inult. 3*902, and 597'i6, 
0nd *03i4728 all together. 

ygo2 its log. iso'59i2873 
597*i6 - - 2*7760907 

•0314728 - - 7-4 979353 
Prod. 73*33533 - - J-8653I33 

The 2 cancels the 2, and the 1 to 
carr^ from the decimals is fet down. 



EA'.2.Tomul.2'58i926by 3*457291. 
2^581926 its log. as 0*41 19438 
5*457291 . - 0J387359 

Prod. 8-92647 - - 0*9506797 

Ex. 4. To mult. 35*86, and 2*1046, 

and 0*8372, and 0*0294 all together. 

3-586 its log. is 0*5546103 

2*1046 • -' 0*3231696 

^ 0*8372 - - ^9228292 

0*0294 - - 2*4683473 

Prod. -1857618 - -"1*2689^64 
Here the 2 to carry cancels the J, aud 
there remains the 1 to fet down. 



II. Divijion by Logarithms, 

From the logarithm of the dividend fubtraS the logarithm of the 
divifor, the remainder is a logarithm whofe correfponding number ' 
will be the quotient required. 

But firft obferve to change the fign of the index of the logstrithm 
of the divifor, viz. from negative to affirmative, or from affirmative 
to negative ; then take the lUm of the indices if they be of the /ame 
kind, or their difference when of different figns, y^ith the fign of 
the greater, for the index to the logarithm of the quotient. 

And when \ is borrowed in the left-hand place of the decimal 
part of the logarithm, add it to the index of the logarthm of the 
divifor when that index is affirmative, but fubtra<5l it when nega- 
tive 5 then let the index thus found be changed, and worked with 
as beforet 



S 3 



Ex^ 



«3* 



DESCRIPTION AND USE 



JEa\ I. To divide 24163 by 4567. 
Divid. 24163 its log. 4*3831509 
Divis. 4567 - - 3'6<; 96310 



Qjot, 5*290782 - 0*7^35^99 

Ex* 3. To divide -063 1 4 byoo7 24 1. 

Divid. •06314,118 log. 2*8003046 

Divis. "007241 - 3*8597985 

Quot. 8*7197(32 - 0-9405061 
Here 1 carried from the decimals to 

the 3 makes it become 2, which taken 

from the other 2, leaves o remainiog. 



Ex. 2. To divide 37*149 by 523*76, 
♦ Divid, 37*149 its log. i'569947i^ 
Divis. 523*76 ' - - 2*7191323 

Qx^ot. '07092752 - 2*8508148 

Ex* 4. To divide '7438 by 12*9476, 

Divid. '7438 its log. i"87i456i 
Divis. 12*9476 - i*ii2t89j 

Quot. '05744694 - 2*7592669 

Here the i taken from the 1 makes 
it become a to fet down. 



III. I'he Rule of Three^ or Frofortion. 

Add the logarithms of the 2d and 3d terms together, and from 
their fum fubtradl the logarithm of the ift, Ky the foregoing rules ; 
the remainder will be the logarithm of the 4th term required. 

Of in any compound proportion whatever, add together the loga* 
rithms of all the terms that are to be multiplied^ and from that fum 
take the fum of the others j the reihainder will be the logarithm of 
the term fought. 

But inftead of fubtracEling any logarithm. We may add its comple-r 
ment, and the refult will be the fame. By the complement is meant 
the logarithm of the reciprocal of the given number, or the re- 
mainder by taking the given logarithm from o, or from lO, changing 
the radix from d to 10; the eafieft method of doing which, is to 
begin at' the left hand, and fubtraft each figure from 9, except the 
lafl fignificant figure on, the right-hand, which muft be fubtraded 
from 10. But when the index is negative, add it to 9, and fubtradl 
the reft as before. And for every complement that is added, fub* 
tra<St 10 from the laft fum of the indices. 



Ex. I . To find a 4th proportional to 

72*34, and 2*519, and 357*4862. 
As 72*34 - com p. log. 8*1406215 
To 2-519 - - 0*4012282 
So 357-4862 - . ^-5532592 

To 12*44827 - . 1*0951089 
Ex. 3. To find a number in propor- 
tion to '379145 as '85132 is to 
'C649 
As *o649 ^ comp. log, 11^*1877553 

To '85132 - - 1-9300928 

So '379145 - - 1*5788054 

To ^•973401 - . . 0*6966535 



Ex. 2. To find a 3d proportional to 

12*796 and 3*24718. 
As 12*796 - comp. log. 8*8929258 
To 3*24718 - - o*Ji'i5o64 
So 3*24718 - - 0*5115064 

To -8240216 - - 7*9159386 

Ex. 4^ If the intered of looL for a 

year or 365 days be 4*5!. what will 

be the interefl of 279*25!. for 274 

days. 

. fiooT , r 8*0000000 

- ^'445993^ 
2*4377506 

0*6;^2I2< 



To|^79-^S 
U74 

So 4*5 

To 9'433296 



««>v 



0*9746634 



' OF THE TABLES. 

IV. Involuthfiy 6r ratjing of Powers^ 



«3I 



Multiply the logarithm of the number given by the propofed 
index of the power, and the produ6l will be the logarithm of the 
power foi^ght. 

Note^ Jn multiplying a logarithm with a negative index by any 
affirmative number, the product will be negative. — But what is to 
be carried from the decimal part of the logarithm will be affirma- 
tive.— Therefore the difference will be the index of the produd; 
and is to be accounted of the fame kind with the greater. 



£;r* !• To find the 2d power of 

^•5791. 
Root 2*5791 its log, 0*4114682 

index " • - ' 2 

Power 6'65i756 - 0*8229364 

]Ex. 3* To find the 4th power of 

•. 9163, ^ 

Root *09i63 its log. 2*9620377 

index - . . 4 

Power •oooo7049i8-J*848i5o8 
Here 4 times the negative index be- 
ing 8, and 3 to carry, the diflfereDce 
J is the index of the produ6^« 



Ex* 2. To find the cube of 3*o7i46« 

Root 3*07146 Its log. 0*4873449 

index - - - ^ 

Power 28-97575 - I '4620347 

Ex, 4. To fin(J the 365th power oif 

1*0045. 
Root 1*0045 ^** ^°g« 0*0319499 

index - - 36 ^ 

9749S 
J 16994 

<;8497 
Power 5*248888 - C7117135 



V. Evolution^ 0r ExtraSfion of Roots. 

Divide the logarithm of the power or given number, by its index, 
and the quotient will be the logarithm of the root required, 

iJotej When the index of the logarithm is negative, and the 
divifor is not exactly contained in it without a remainder, increafe 
it by fuch a number as will make it exadly divifible ; and carry the 
units borrowed, as fo many tens, to the left-hand place of the de- 
cimal part of the logarithm i then divide the refults by the index of 
Jhc root. 



Ex^ 



»34 



DESCRIPTION AND USE 



Ex* I* To find the fquare root of 

365. 
Power 365 - - 2)3*5622929 
Root 19*10498 - - 1*2811465 

Ex* 3. To find the lotb root of 2* 
Power 2 - - io)o'30io30o 
Root i'07i773 - - 0*0501030 



Ex* 5* To find the fquare root of 

•093, 
X Power '093 - - 2) 2*9684829 

Roof 304959 • - I -48424 1 5 
Here the divifor 2 is contained ex- 

a£^ly once in 2 the negative index, 
therefore the index of the quotient 

is If 



Ex» 2. To find the cube root of 

12345. 
Power 12345 - - 3)4*o9i49ii 
Root 23.1 1162 - - i'3638304 

Ex* 4« To find the 365th root of 

1-045. , 
Power 1*045 -- 365)0*0191163 
Root I'ocoizi * - 0*0000524 

£^. 6. To find the cube root of 

•00048. 

Powe? . • , 3)4*68 H4 1 2 

Roofo7829735 - - 2*8937471 

Here the divifor 3 not being exadly 

contained in 4, augment it by 2, to 

make it become 6/ in which the cii- 

vifor is contained juft 2 times ; and 
the 2 borrbwed being carried to the 
other figures 6 &c, makes 2*681 2412, 
which divided by 3 gives •8937471. 



OF THE TABLES FOR LOGARITHMS TO TWENTY 

PLACES. 

THESE are tables zd, 3d, and 4th, beginning a^ page 187. 
Of thefe, table 2 contains all numbers from i to looo, and 
all uneven numbers from 1000 to 1 161 ; with their logarithms to 
twenty places : table 3 contains all numbers f^om loiooo to 
101139, with their logarithms to twenty places, and the ift, 2d, 
and 3d differences of thofe logarithms : apd table 4 contains all 
logarithms regularly from ooooi 1090139, with their correfponding 
natural numbers to twenty places, as alfo the ift, 2d, and 3d dif- 
ferences of thofe numbers. And by means of them may be found 
the logarithm to any other number, and the number to any other 
logarithm, to twenty places of figures* 

(L) To find thi Logarithms to given Numtirs. 

Case i. If the given number 3 be found in any of thefe three 
tables ; then its logarithm B is in the line even with it. 

Case 2. If b is known to be the produft or quotient of nunrbers 
found in thefe tables ; then B is the fum or difference of the loga- 
rithnas of thofe numbers. 

Case 



OFTHETABLES. 135 

Case 3. If a\ the firft fix fignificant figures of a given number 
Vy be found in table 3 ; let a^ be an integer. A' its logarithm ; 
J the remaining figures oi hf \ x the complement of ^ to ^' or i j 
D', D^', D''% the ift, 2d, 3d di fferen ces of the logarithms in the 

fame line with A' ; /= \ D'^^ x ;r + i'+ D^' ; Then B' the loga- 
rithm of the number hf will be 

D^ X ^ + A^ - - to lap 
4 X D^' + D' X ^ + A/ - - to 1 7>placcs of figures nearly, 
kx.f +D'x ^ + A' • - to 20-) 
Ex, I. Given the number i' =: 0-01010,26227,6351, to find B' its 
logarithm nearly to twelve places. 

Heretf' = 101026. A' = oo443>3i579»747 
I = 0-2276351 ^ D' +' 9785,6 1 8-^ 

ly = 429881746 B' = 2-00443,41 36 c,365— 

Ex, 2. Given h^ =z o*oibio,26227,63509,626, to find B' its Jog. 

nearly to 17 places. Here cl = 101026. 

J = 022763,509626 i;r = 0-772365 J D' = 42988,174579; D'^ 

= 425510. 

Now ixW • - - - 16432,45 

D^ . . - 429 88,17457.86 

X ;r P/^ + D^ - - - 42988,33890131 

k xW + D' X 3^ - - - 9785565466,42 

A' - 00443,31579,74695,33 

And i ;r D'' + D' X ^ + A', or B> 2-00443,41365,40161,75 
Ex. 3. Given b^ = 001010,26227,63509,62573,17345, to find B/ 
Its log. nearly to 20 places. a^ zz ioioz6. 

^ =0-22763,50962,573173; :r =077236,490374; x+i=i'77236s- 
1)^=42988.17457,8 6301 ;D'^= 42550,96343 J IV =84236. 

!Now i ly^^ X X + I - - - - 49766 

D/' - - - 42550,96343 
f 42551,46109 

f'/ _, ^^16432,62757 

D' - - - 42988,17457,86301 

l_£/+jy . . . 42988,33890,49058 

|;r/+ D' X » - - - 9785^65466,45604 

A' - - QQ443>3i579>74695>.^279i 

And B' - - - 2>00443>4i365,40i6i,78395 

Case 4. If the number b come under none of the preceding 

cafes: put^ for the firft five figures of ^j « for loi, the leaft, or 

fome one, of the numbers in table 3 j then— or ^= ^ is to be 

n a ' 

had in table 2, with A its logarithm ; let i^ = — or ba, and i/ 
the firft fix fignificant figures of y (found in table 3) be an inte- 



« • 



ij6 DESCRIPTION AND USE 

ger, and A' its logarithm ; put > for the remaining figures of i' ; 
5f the complement of * to ^'; D^ D'', D'^', the lit, 2d, 3d, dif- 
ferences of the logarithms in the fame line with h! \ f "zz ^ D'^' X 
IT+T + D". Then B the logarithm of the number b will be 
D^ X ^ + A' ± A = B' tAto 12I 

^^^O^' + D' X ^ + A/ ± A =: B' ± A to 17 )^Ce?ne^^^ 
I ;, / + D^ X » + A' ± A = B' ± A to 20 J ^2"'''' "^^''^^ 

Ex. Given * = 3' H' 59.^6535,89793,23846,26434, to find B to 
twenty places. 

Here a == 31415- Let j = ^ =: 31 1. 

b 

Then A^ = - = o 01010,15840,9514^,02970,57 ; <?' =: 101015. 

a 

i = 0-84095, i440^»97oS7 > ^ =o*^S904>8s597 ; * + x = i'i5905 5 
D'=: 42992,85574^06337; iy^= 41560,23099; D//^ := 84263. 

Now ^ D'" X * + I - - - - 3^555 

» D'^ - - 42560,23099 

/ - - - 7 42560,556^4. 

ixf - - - - 33«4.5976i 

D' - - - 42992,85574^063-^7 

I xf+ D^ - - - 42992,88958,66098 

|;r/+D^x i - - " 36i54i93242,039«9 

A' - - 00438,58681,74054,30961 

A - 49g76>03890,268.'^7»5055^ 
AndB -. - 0-497 14,98726,941 33,85435 

Or, let fl = - = 3*^«6 = 0^536 x 6. 

Then A' =: Aa = 10-103^36, 1 97 39,44? 7 5^0549; a' = 1010^3. 
» =o-.6i973,94477»50549; ^ =0-38026,055225 ;x+i=z 1-38026; 

iVrr 42?8^, 1 961 8,80760 ; W Z= 42545,06747 ; D''' =84219. 

Now t ly^' X x+ 1 - - - - 38748 

D'' - - - 4gS45>o6747 

/ - - - - - 4^S45i4S495 

^xf 8089,17910 

D' - - - 42985,19618,80760 

|y/+ D^ . ^ - 42985,27707,98670 

Ixf+Uxi - - - a6629,67i«7,888ii 

A' - 00446,32488,03359,61 854 

B' - - - i'00446,59i27,70547,50665 

A - - 0*5073 f ,60400,7641 3,65230 

B = B'— A - - 0-49714,98726,94133,85435 

(II.) To fin J the Numbers iogivin Logarithms. 

Case i. When the logarithm B is found in any of thcfe three 
tables : then its number b is in the line even with it. 



OF THE TABLES. 



»37 



Case a. If the fii-ft five figures (omitting the index) of a given 
logarithm B', be between OO432 and 00492 : take them as an inte- 
ger, and put A' and C for the logarithms, in table 3, next lefsand 
greater than B\ a^ and / their numbers j let D' {= C — A') and 
D'' be the ift and 2d differences in the line with A'j A = B' — A'; 



+ >, nearly true to 17 places of figures. 

. f'L 9i'y^'^ thejogarithm B' - - = 5*00446,59^27,705411107 

to find y Its number. A' = 5,o0446,32488,O3't^s^9 

fl' = 101033 ^ - 0-26639,67187,888 

9 - - ■ 0-6197^^,044776 D' = 0-42 985,19618,808 

*' = '^i<^33'6i973>944776 D' — a = 0-16345,52430,920 

X s= 0-38026 • 
D" zr o«ooooo,42545 
J X D" =: 0*00000,08089,1 
„ ^ D' + f X D" = 0-42985,27707,9 

But when any other logarithm B is given, fubdu£i '004321 from 
the firft fix figures of B ; call the remainder R, and let A be the 
logarithm in table 2, next lefs than R, or next greater than the 
complement of R, and a its number : then B' =: B — A, orB' = 
B -f A, will be within the limits of table 3, and *' will-be found 
as in the preceding example ; and if B' = B — A, then i zz at' i 

orifB/=?B + A, then*= i\ 

a 

Case 3. If A', the firft five figures (omitting the index) of a 
given ^logarithm B', be found in table 4 : let a' be its number; and 
put A as an integer, and 4 the remaining figures of B', and X the 
complement of A toD'j A A d'", the ift, 2d, 3d diff erences o f 
the numbers in the fame line with a' j /= /'— f d"' x X + 1 : 
^hen the number A', whofe logarithm is B^ will be 

d ' X A + a' to 12 J 

d' — kX d' ' X A + fi' - - - to 17 ( places of figures nearly. 

d^— Xf X A + a' to 203 

Ex. Given the logarithm B' zr 000006,93311,37711,69929, to 
find h' its number to 20 places. Here A' = 00006. 

A = 0-9331 1,37711,69929 ; X =0-06688,622883; X+X=: I •0668865 

V z: 23029,29742,ti293; ^'= S30^M^746 ; <r" 2:1,22100. 



Now 



'k 






Now 


itf" X 


X + i - 


f 


|X/ 


9m 




fX/ 


• 

A 

of ■ 



138 DESCRIPTION AND USE 

- - - 43422 

550^7>52746 

53027,09324 
T 1773*39* 15 

23029»2974.2,2I293 

23029,27968,82178 
21488,93801,72000 
- ioooi,38i64>64943*57474- 
And*' - r r i '00015,965 35,874S2>9474 

Case 4. If the logarithm B come under none of the preceding 

• cafes. Put A for the logarithm in table 2, next lefs than B, pr next 
greater than the complement of B, and a its number ; let B' = B 
— A, or B' = B + A i and A', the firft five figures of B', may be 
had in table 4, with a^ its number ; put A' as an integer, and let a 
be the remaining figures of B' ; X the complement of A to D' ; i', iff^^ 

* d'^^j the ift, ad, 3d differences of the numbers in the fame line with 
^l f=df^ — ^ d'^' X X+ I : then the number *', whofe logarithn:| 
16 B', wil^ be ._ 

■ ■ ^^ , ■ ,. ^ X I places 'of ngurc$ 

df^jXd^f XA^TV xa = aytoi6J J^^^j^^ ^ 

df^iXf X A + ^ X J = ai' to 19J 
Ex. Given B =:'4'46372,6i 172,07 184,15204, to find b its number. 
' Let A = r46239*79978*98956>o8733. a = 29. 

B' = B— A zss'-oo 1 32,8 1 1 93,08228,06471'. A' = 00132. 

A =r 0*8 1 193,08228,06471 ; X =0'i88o6,9i772 ; .X+ i = 1-18807 | 

^=z i3096;2o835.34S^^' = S3>8i,S973}; ^"^ =1-22457. 

Now i d'f^ X X + » - - ' 48496 

4ff . . - . - 53'8i>597^3 

/ - - T - - 53^81,11237 

fX/ • - - r . 5000,86402 
J/ - - - - 23096,20835,34589 

f — i Xf - - - 23096,15834,48187 

V— iX/x A - - - 18752,45284,85771 

fl' - 10030,44036,01963,96855 

Jf - ^ *. - • 10030,62788,50248, 82626 

hzzaV - 000029,08882,08665,72159,6154 






OP THE TABLES. 



[39 



Or, given B = 4-46372,61 i72,07i84,iS204, to find*. 

^J^^ = 2-53655,84425,71530,11205, ^=344. 

B' = B +. A =1-00028,45597,78714,26409. A'= -00028. 

A =0-45597,78714,26409; X = 0-54402,21286 ; X + I = 1-54402 ; 

^= ^304o,9^^9»9«S^i[y^ 53054,39634; i^// =: 1-22163. 

Now t dff^ ^ XT"i - . . 62874 

^" * • - - S3Q54.39634 

/----- 53^53*76760 

l^f - « ^ i443i»2"79 

, ^ - ' - - - _2304O^9662 9^91521 

^JzULf * - • - 23040,82198,7034^ 
^' — jX/XA - - » 10506,10496,55627 

^ . - 10006,44931,70511,67281 

* " " - 10006 ,55437,81008,22908 

z *' — 7 

*=:^ - • 000029,08882,08665,72159,616 



OF THE TABLES FOR LOGARITHMS TO SIXTY- 
ONE PLACES. 

Thefe are tables 5 and 6, from page 204 to page 208 ; the former 
containing the natural numbers in regular order from i to loo, and 
after that all the primes up to iioo, with their ^correfponding loga- 
rithms to fixty-one places of figures ; and the latter in page 208 
contains all numbers in order from 999980 to 1000020, with their 
logarithms to fixty-one places, as alfo the ift, 2d, 3d, and 4th 
diiferences of thefe logarithms. And the ufe of thefe tables, in 
finding the logarithm to any number, or the number to any loga* 
rithm> each to fixty-one places of figures, will be as follows. 



T a . I. J99 






ts 



,3-3 t'Se-: 

° lilt 

■: s S°^ =^? S- =^*s=? 
^ i •|'5! :§.:■ '5^ "'S-liifS 

•^iss 'IJ-li-^jS Sf infill 

IpllUsllllill till 

§u«c*^^3 2 wo =^>£. SaS "^w " 

> X a 2 S s, fJ a.^ ° EP.S "32 r:'" I-" 
^•^•g 5 S S i « V «-t > "^"-S 3-.^5 S 

^ tS „-B '^^ZZ,^^ E •''- " ^8 *= 
^,-2 ?-s e^ -^-m:: fe-^ ViK'o 2'8"°^ 

kUi !ii|it3lli1|- 

iiii! itf^-:-jiiii-ii-i ■ -'' 

•aJi'^S I I'll 1 l-3.s-^i|,^1 - iJ'» 



s .a 



Egg" 

V Ssf.s-s 



■B jjo 



3 






1^4'^ J.! 






^:5| 






II 



I 5,! 

J ^ J 

1 1^ 









fSlSf-j 









St :s?»- 



ii=-sli 



^^1 






to t>iK0.6**«'^ 






SS3;"- 



«««<< 






^■'"i 



l!|tli 



I 



8 J! a" 



SsJa 






^:.sg 






1 3 rzll 

i-ilAlil. 



OFTHETABLES. Ml 



OF THE TABLE OF HYPERBOLIC LOGARITHMS, 

This is table 7, in pages 209 - - - 212, which contain the 
libries of numbers I'oi, i*02, 1*03, &c to lO'Oo, with their hy- 
perbolic logarithms to feven places of figuj^s. They are fo called 
becaufe they fquare the afymptotic fpaces of the right-angled hyper- 
bola ; and they are very ufeful in finding fluents, and the fums of 
infinite feries* The table» as well as the following rules, were firft 
given at the end of Simpfon's fluxions ; but they were rcndere4 
much more corred in the. French edition of Gardiner's tables, 
printed at Avignon in 1770, being very incorreft in the laft figure 
in Simpfon's book. But both thofe books are very erroneous ia 
|he example for finding logarithms by the table. 

I. fFhen the given Number is between % and 10. 

From the given number fubtrad the next lefs tabular number, 
divide the remainder by the faid tabular number increafed by half 
the remainder ; add the quotient to the logarithm of the (aid tabular 
number, and the fum will be the logarithm of the number pro- 
pofed. 

Ex. To find the hyperbolic logar 
rithm of 3-45678. Here the fiext lefs 3'4S339)'Oo678(-ooi9633 
number is 3*45, and its logarithm ^'^38.^74^ 

I •23?3742, the remainder or dividend log. l •2403375 

•PO678, its half 339, which joined 

to the tabular number 3*45 gives the divifor : the quotient 
*ooi9633 added to the tabular logarithm 1*2383742^ gives 1*2403375 
^he required logarithm of 3*45678. 

2* When the given Number exceeds 10. 

Find the logarithm of the number as above, fuppoiing all the 
figures after the firfl to be decimals ; then to that .logarithm add 
^^•3025851, or 4*6o5i*702, or 6'9077553, &c, according as the 
given number contains 2, or 3, or 4, &c, places of integers, 
That is, add 2*302585092994 multiplied by the index of the power 
of 10, by which the given number was divided to br^g it to one 
integer, or within the limits of the table. 

Ex. To find the hyperbolic logarithm of 345*678. 
This number divided by 100 or lOS to bring it i '2403375 
within the limits of the table, or removing the deci- 4*605 170Z 
mal point two places, gives 3*45678, the logarithm 5*8455077 
of which as above found is 1*2403375, to which add* • 
Ing 4*6051702 thehyperbolic logarithm of loO, thefum i* 5*8455077 
^he hyperbolic logarithm required of 345*678* 



144 DESCRIPTION AND USE 

NoUy The hyperbolic logarithm of any number maybe alfo found 
from Briggs's logarithms, viz. multiplying Briggs*s logarithm of 
the fame number by the hyperbolic logarithm of lo, viz. 
Multiplying by - ^ a -30258,50929,94045,68401,799 14, 
Or dividing by its reciprocal '43429)448 1 9,0325 1,82765,1 1289. 

OF^THE LOGISTIC LOGARITHMS. 

Thcfe are in table 8 pages 213 - * • 217, which contain the 
logiftic logarithm of every fecond as far as the firft 80' or 4800^. 

The logiftic logarithm of any number of feconds, is the difpe- 
tence between the logarithm of 3600'' and the logarithm of that . 
number of feconds. 

The chief ufe of the table of logiftic logarithms, is for the ready 
computing a proportional part in minutes and feconds, when two 
, terms of the proportion are minutes and feconds, hours and minutes^ 
or other numbers. 

• When two terms of the proportion are common numbers, their 
common logarithms may be.ufed inftead of their logiftic logarithms, 
putting the logarithm where its complement (hould be, and the 
contrary. 

I. Tofindtbi Logiftic Logarithm of any Number of Minutes and St* 

condsj within the Limits of the Table. 

• 

At the tbp of the table find the minutes, and in the fame co- 
lumn^ even with the feconds on the left-hand fide, is the logiftic 
logarithm. 

Notey When hours are made any terms of the proportion, they 
are to be taken as if they were minutes, and the minutes of an hour 
as if they were feconds. 

2* To find the Logiftic Logarithm of any Number not exceeding 4800. 

In the 2d row next the top of the table find the number next lefs 
than that given, and in the fame column, even with the difference 
on the left-hand fide, is found the logiftic logarithm. 

When two giyen terms of the proportion are common numbers^ 
pne or both greater than 4800, take their halves, thirds, &c, in- 
ftead of them. But when only one of the given terms is a common 
number, and that greater than 4800, take its half, third, &c, and 
multiply the 4th term by 2, 3, &c. 

Irhe logiftic logarithms in this table are all affirmative, as well 
above as below 60' ; but the index of thofe above 60^ is — i ; be- 
low 60' down to 6'> the in^ex is o; and below 6', the indices 
(being either i, 2^ or 3) are cxprefied in the table. 

EXAMPLES. 



OF THE TABLES. 



H5 



EXAMPLES. 



As M 

To 46' 12^' 

So 8 7 
To 6 15 



lo; log. 
o«ii35 
o*8688 



As 60' 



lo. log. I As 60' 



0*9823 



As 46' 12'' .CO. 1*8865 
To 60 o ./ . 0*0000 
So 6 15 • . 0*9623 

To 8 7 . . 0-8688 

As 60' • • CO. ©•oooo 
T04721 . . 1*8823 
So 37' 28" . 0*204$ 

To 2948 • • 



AS DO 10. tog* I AS UO 

T078' 27'' . . 7*8836 To 1 53 1 



So 13 
T018 



S3 
9 



0*6357 



0*5193 



As 4721 
To6o'o" 
So 2948 

To 37' 28 



. 0*0868 

CO. 0*1177 

• 0*0000 
. 0*0868 

• 0*2045 



As 78' 27" . €0.0*1164 
T060 o . . c*oooo 
So 18 9 • . 0*5193 

T013 53 . . 0^6^ 

As 24b • • CO. I *6o2 1 
T046' 11" . . 0*1137 
So 8h 7' . . 0*8688 

To 1 5' 37'-^ . . 0*5846 

As 46' 11''' . CO. 1*8863 
To24'» • . . 0*3979 
So 15' 37" . 0*5846 

To 8»> f . 0-8688 



So 40 
T01179 



12'' • 



lo. log« 

0*3713 

o'"5S 
0*4848 



As 40' 12'' . CO. 7*886$ 
Toii?9 • • 0*4848 
So 60 o'^ . • o*oooo 

T01531 . , 0*37x3 



As 24^ 4 

To 76' 34" 
Soi3'> 53' 

T044' 17" 

As 76' 34" 
To 241* . 
So 44' 17'^ 

Toi^»» 53' 



• CO. I*602Z 

1*8941 

• 0*1319 

• CO. 0*1059 

• • o*3979 

• 0*1319 

• ^'^357 



The logiftic logarithms may conveniently be ufed in trigonome- 
trical operations, when two of the terms are fmall arcs, with the 
logarithmic fines or tangents of other arcs : obferving, that inftead 
of the logarithmic fine or tangent, to take the complement of their 
logiftic logarithm ; and the contrary. 

But this may be as readily and more naturally done by the loga- 
rithmic fines and tangents themfelves of fuch fmall arcs, as. taken 
from the next following table of fines and tangents for every fecond 
of the firft 2® or 120'. 



OF THE LOGARITHMIC SINES AND TANGENTS TO 

EVERY SECOND. 

Table 9, pages 218. - - - 247) contains the log. fines and 
tangents for every fingle fecond of the firft 2 degrees of the qua- 
drant ; the fines being placed on the left'-hand pages, and tangents 
on the right. The degrees and minutes are placed at the top of the 
columns, and the feconds on the left-hand fide, of each page, the 
logarithmic fine or tangent being found in the common angle of meet- 
ing. So of 1^52^54^^ the log. fine is 8*51634201 and the log. tangent 
8*5^65762. 

The fame numbers are alfo the cofines ;ind cotangents of the laft 
2 degrees of the quadrant^ thofe degrees with their minuter being 
placed at the bottom of the columns, and their feconds afcending 

U oa 



146 DESCJIIPTION AND USE 

on the right-hand fide of the pages. So the cofine of 88^ 7' f/' \% 
8'5i63420, and its cotangent 8-5165762. 

. When itfe required to find the fine or tangent &c. to 3ds &c. or 
any other fradional part of a fecond« fubtradl the tabular fine or 
tangent of the compleat feconds from the next tq it, in the table, 
and take the like proportional part of the difference ; which part 
added to, or taken from, the (aid tabular fine or tangent, accord- 
ing as it is increafing or decreafing, will give the fine or tangent 
required. 

i:*.Tofindthelog.fineof 1° S^' S4"^5'" or ^"^ 5*' S4" \\ or A. 
Here thefineof 1° 52' 54'' 1° S^' 54'^ fine 8-5163420 

taken from the next leaves ^ ^^ ii * \ * 8^164061^ 

641, which multiplied by dif. 641 

5 and divided by 12, or 5 

multiplied by 25 and di- 12)320$ 

vided by 60, gives 267 the pro. part 267 

pro. part.; this added to 1^,52' 54" . , 8'$i6342o 

the firft fine gives that 1° 52' 54'' 25'"8'5i63687 

which was required. 

On the contrary, if a fine or tangent be given, to find the cor- 
refponding arc \ take the difference between it and the next lefs ta- 
bular number, and the difference between the next Iei*s and greater 
tabular numbers, fo fbali the lefs difference be the numerator, and 
the greater the denominator, of the fradional part to be added to 
the arc of the lefs tabular number ; which fradlion may alio, if re- 
quired, be either turned into a decimal, or into 3ds &c. by multi- 
plying the numerator by 60 and dividing by the (^nominator. 
Ex. To find the arc whofe fine is 8'5i639CX>. 

Finding the number is between the 
.fines of 1° 52' 55" and i*' 52' 54^ i"" 52' 55'' - - - 8-5164061 
take the differences between the fines ^ S^ S4 - - - - 8*5163420 
as in the margin, and the differences ^ 5^ 54 4S" ' 8*5 163909 
'give -IIt for the fraSion of a fecond, diff. - ^ 480 

or I J nearly, which abbreviates to ^^^ - - 641 

— = 45'^' 5 and therefore the arc fought is i*" 52' 54'^ 45'". 

Where the ift differences of the fines and tangents alter much, 
as near the beginning of the table, the 2d, 3d, &c. differences may 
be taken in, and then the logarithmic fine or tangent will be exprefTed 
by this ferie$, viz. 

Q= A+;riy+;r-^ D^'+at. i=i .iZ:?D'" &c, or nearly A+D'— ID":^; 

where A is the next left tabular logarithm, IK, D", D% &c. the 
ifl, 2d, 3d, &c. differences of the tabular logarithms, and x tHe* 
fractional part of the arc over the compleat feconds. 



Ex. 



O F T H E T A B L E S. 147 



, _ , , ly and the mean id diff. D^' =:-: — 48. Hence 

14404 ^^ 
^4357 



A : 

xjy . 


m ^ 


?• 


1641417 

r 2977 


X — ■ 

X, — 

z 


!d" 


- 


- - 4 



£;r. To find the log. tangent of 5' i" 12''' i^^^'or 5^. i '' ^V?5 ot 5* i"'306. 
Tang. T^. HcreA=7-i64i4i7;Ar=3%\;D'z=i4404; 

5^0''- - 7-1626964 ^ '^^ ^^ ..«.T.,/_ ,0 rT._„ 

5 I - - 7-1641417 

5 2 - - 7-1655821 ,77^; -47 

S3 - ■ 7'i67oi78 

Thereforeihetangcnt of 5' i'' 12'''' 24" - - - - 7-1644398 
And on the other hand, when t}>e fine or tangent is given, and 
falls near the beginning of the table, from the fame feries we may 
iind X the fra£iional part of a fecond. For fuppofe it be required 
to find the arc whofe tangent is 7-1644398. This falling between 
the tangents of 5^ i'' and 5' 2'', take the differences &c. as above, 

and the feries gives 7"i644398 =: 7-1641417 + xW + x. EK"^, 



or 2981 = 14404*' — 24..** — Xy or — 24;r* + 14428;^= 2981 5 
which givfes *= -2067 '' nearly = 12''' 24'*^ Therefore the arc 
required is 5' 1" 12''' 24''. Or rather the approximate value A + 

D -iD .x-Q, g.ves*-.jy— ^-^^^^^_^^^^^-— - 

•2067, the fame as before. 

m 

OF THE TABLE OF NATURAL AND LOGARITHMIC 

SINES, TANGENTS, &c. 

Table 10, page24& 337> contains all the fines, tangents, 

fecants, and verfed fines, both natural and logarithmic, to every 
minute of the quadrant, the degrees at top, and mrnutes defcending 
down the left-hand fide as far as 45°, or the middle of the quadrant, 
and from thence returning with the degrees at the bottom, and the 
minutes afcendingby the right-hand fide 1090°, or the other half of 
the quadrant, infuch fort that any arc on the one fide is on the 
fame line with its complement on the other fide ; the refpcftive 
fines, cofines, tangents, cotangents, &c. being on the fame line 
with the minutes, and in the columns figned with their refpec- 
tivfc nanies, at top when the degrees are at top, but at the bottom 
when the degrees are at the bottom. The natural fines, tangents, 
&c, are placed all together on the left hand pages, and the loga- 
rithmic ones all together, facing them, on the right-hand pages. 
Alfo in the naturals there are two columns of common difl^erences, 
and in the logarithmic 3 columns of common differences, each co- 
lumn of dilFercnces beinj/ placed between the two columns of num* 
bers having the fame differences ; fo that thefe differences ferve for 
both their right-hand and left-hand adjacent columns : alfo each 
differential nutnber is fet oppofite the fpace between the numbers 
whofe difference it is. The numbers on the fame line in thofe co- 
lumns having fqch common differences, are mutually complements 

U Z of 



I4« DESCRIPTION AND USE 

of each other; fo that the Aim of the decimal figures of any two 
fuch numbers^ is al^rays i integer, with o in each place of deci^ 
mals. 

All this will be evident by infpcding one page of each fort, sis 
well as the method of taking out the fine, &c. to any degrees and 
coniplcat minutes. It is however to be obferved, that in all the 
Jog. fmes, tangents, &c. and in fuch of the natural as have any 
iignificant figure for their index or charadVeriftic, the indices are 
exprefled in the table, and the feparating point is placed betweea 
the index and the decimal part of the number; but in ieveral co- 
lumns of the natural fines, &c. having o for their integer or index, 
both the index and decimal feparating point are omitted : and where- 
cver this is the cafe, it is to be undcrfiood that all the figures in fuch 
columns are decimals, wanting before them only the ieparating 
point and index o. 

The fine, tangent, or fecant of any arc, has the fame value, or 
is exprefled by the fame number, as the fine, tangent, or fecant of 
the fupplement of that arc ; for which reafon the tables are carried 
only to a quadrant or 90 degrees. So that when an arc is greater 
than go% fubtrad it from |8o% and take the fine, tang, or fecant of 
the remainder, for that of the arc given* But this property does not 
take place between the verfed fines of arcs and their fupplements : 
and to find the verfed fine of an arc greater than 90% proceed thus : 
in the natural verfed fines, to radius add the natural cofine, the fum 
will be the natural verfed fine ; and in the Ipg. verfed fines, adc^ 
0*3010300 to twice the log, fine of half the arc, the fumj> abating 
radius ip'ooooooo, will be the log. verfed fine required. 

X. GiveninPfArc\ to find its Sim^ Cajhuy Tangent^ ^c. 

Seek the degrees at the top or bottom, and the minutes refpeftive^ 
ly on the left or right, and on the fame line with thefe is the 
fine, &c. each in its proper column^ the tide being at the top or 
bottom, according as the degrees are. 

fiut when the given arc contains any parts of a minute, interme- 
diate to thofe found in the table: take the difference between the 
tabular fines, &c. of the given degrees and minutes, and of the mir 
nute next greater ; then take the proportional part of that difiFerence 
for the parts of the minute, and add it to the fine, tangent, fecant, 
and verfed fine, or fubtr^ift it from the cofine, cotangent, cofecant, 
or coverfed fine,^ of the given degrees and minutes; to ihall the fun^ 
or remainder be the fine, &c. required. 

Notiy The proportional part is found thus, as 1' is to the given 
intermediate part of a minute, fo is the whole difierence, to the pro- 
portional part required ; which therefore is found by multiplying 
the difference by the faid intermediate part. Alfo that intermediate 
part may be expreflied either by a yulgar fraSion, or decimal, or a 
fexagcfimal in feconds, thirds^ ^c. an^ the fjra^ion or fexagefimal 

may 



OF THE TABLES. 



^49 



may be firft reduced to a decimal, if it be thought better fo to io^ 
by dividing the numerator of the fradion by the denominator, or by 
dividing; the fi^xagefimai by 60. 

'examples. 



1. To find the natural fine of 



" 28'' 12'". 



1^48 

Id the column of difference between 
jhc natural fines of 1* 48^ and 1^ 49' 
is the difference 2907 ; and 28'' 12'" 
being = 28*2''' = -47' ; therefore as 
1 : 2907 : : '47 : the pro. part 4- 1366 
to which add fin. 1* 48'. - 0314108 

makes fin. of 1 *» 48'' 28'' i a"^ 0315474 

A. To find the nat. coverfed fine of 
4<^ 6' s'' 40'"^ 

1 : 2902 (tab. dif.) 1:5^^=1 ,7^ 

5'' 40"^ : piro. part - - J 
4<> 6' coverf. - * 9285026 

40 6' s- 



40 



/ff 



9284752 



J. To find the log. fee. of 7** 1 2' 50'^ 

^:i6otab.dif. ::>•=: 50": pr.pt.+ 133 

7^i2^fecant « 10*0034381 



7*>i2^5o". - 10*0034514 



2. To find the natural tangent of 
S^ 9' 10'' 24'^^ 
8^ 10' tang. - - 1435084 
8 9 - - - - ^432115 

diff. 2969 

i:2969::(io''24'''==)'i7'f: +515 

• - 143211; 
9' 19" 24''' - 1432630 



8-9 
8<> 



4* To fine the logarithmic coGne of 
6*> 8^42'^ 

I : i36(tab.dif.)::7'=42'': pr.pt. - oj 
6" 8' cofme - - 9*9975069 

t«8'42" - . 9*9974974 



6. To find the logarithm cotaneentof 

1 : 2581 tab^ dif. : : *2oi= > 

i2"2o'":pro.panf ""iS* 

10*0905978 

100905447 



39" 4 
39** 



cotan. 
ao' 



Via"-'" 



The foregoing method of finding the proportional part of the ta- 
bular difference^ to be added or fubtraded, by one fingle propor- 
tion, is only true when thofe diflTerences are nearly equal, and may 
do for all e^^cept for the tangents and fecai^ts pf great arcs near 
the end of the quadrant in the natural fines^ &c. and in the log. 
fines, &c. Except the fines and verfed fines of fmall arcs, the tan- 
gents of both large and fmall arcs, and the fecants of large arcs* 
And when much accuracy is required, thefe excepted parts may be 
fou^d by the feries ufed in the lail article, viz. Q^=s A + ^ D -f* 

x.^^^'^D'^'lfc, or = A + D' — ^D^'x ncar^ 



jr— I 



.D''+ 



2 ; 2 3 

Jyj where A is the tabular number for the degrees and minutes^ 
jy, Ty\ D''\ &c. the I ft, 2d, 3d, &c. tabular differences, and x 
the fra£lional part over the compleat minutes, &c ; at leaft it may 
be proper to find the tang, and fecants of very large arc^ froni this 
feries ; but as to the log. fines, verfed fines, and tangents of fmall arcs^ 
they may alfo be found, perhaps eafier, from their correfponding 
patufa) ones, viz. find the natural fine^ reffed iine^ or tangent qf 

the 



15^ 



DESCRIPTION AND USE 



the given fmall ar(r, and then iind the log. of fuch nttural number 
by the ift or large table of logarithms, which will be the log. fine, 
&c. required. And the log. tangent and fecant of large arcs will 
be alfo found by taking the difference between 20 and their log. 
cotangent and cofine fefpediirely. And ladly, the natural tangents 
and fecants of large arcs may alfo be found by firft finding their log, 
tangent and fecant, ahd then finding the correfpopding number. 

EXAMPtES. 



// ,^///^ 



X. Toimd the log. fine of i* 48' 28 

The itacural fii^e, found in Ex. i • above, is 
*03 1 5474 ; and the log. of this is 8*4989636 
the log. iioe required. 



*^m 



/// 



3. To fibd the log. tang, of a^ 23' 33'' 36 

2^ 23 'its nat. tail. . - - 0416210 

1 : 2914 tab.dif. : : '5^6'zz33" 36'''j-f '632 

2? 23' 33" 36'" nat. tan. 0417842 

Itslog.a aj 33 36 log. tan. 8-6210121 

5. To find the log. fee. of 88^ 1 1' 3 1'' 48'''. 
Its complement is - • i 48 28 1 2 
Its log. £ne in Ex. i. it . • 8*4989636 
Which taken from • • - 2p'ooooooo 
Leaves 1. fee. 88** n' 5^*48"' 11-5010364 



In the 6rh example the natural fecant is 
found by thedifferentiat feries to be 3 1 '698 3 39. 
But by taking the number to the logarithm 
of it, as found lA the 5th example, it is 
3 1 '6983 3 J ; which feems to be the more ac- 
curate, as wellaatheealier way; and itnleed 
this method by the feries, feems to be, in 
feme inftances, more troublefome, and leis 
accurate, than finding the fecant by di\ Ming 
^by tb&cafm43« 



2. I'o find the Jog. vers, of 1^ 48' iSf^ 12 

i^ 48' nar. vers. - - 0004934 
1 : 92 tab. dif* : : -47' =: 28'' 12'^' ; + 43 

1° 48' 28'' 12''' nat. vers, '0004977 
Its log. I 48 28 12 log. vers, • 6*6969676 

4. To find the log. tang, of 87^ 36' 26 ' 24"'. 
Irs complement is • • ^ 23 33 36 
Whofe log. iftng. in Ex. 3. is - 8*6210121 
Taken from -----. 20*0000000 

Leaves log. tan. 87** 36' 26'' 24''' 11*3789879 



6. To find the nat. fee. of 88® 1 1' 31" 
nat. fee. 



88° 

88 
88 
88 
88 



9 
10 

12 
13 



30*976074 

31-^57577 
31-544246 

3r83622S 
32*133663 



48 



//' 



D' 

281503 
286669 
291979 
297438 



D" 

5166 

5310 

54591 



144 
149 



Hence A = 31*544246; D'z: 301979; 
Y>" = 5^10 1 the mean D"' = 146 ; 

^ = *53' = 3i''48'";^.^'=-. -12455; 

A" — I X — 2 ^ 
X. • = '06125. 



Then A - - 

atD' - 

x^i 






31-544246 
- 154748 

. -664 



•1 ;r— 2 



X. 



D'" - . - 



- - 9 



31*698339 



2* Given 



OF ,TH E TABLES. 
2. GivtH any SIh*^ Tangent, He, tt find its Art 



»s« 



Take the diiFerence between the next lefs and greater tabular 
numbers of the fame kind, and the difference between the given 
number and the faid next lefs or next greater tabular number, ac« 
cording as the given number is a fine, tangent, &c» or a cofine, 
cotangent, &c. noting its degrees and minutes ; then the two dif- 
ferences will be the terms of a vulgar fraction of a minute, to be added 
to thofe minutes, to give the arc required. 

And this vulgar fraAion may alfo, if required, be reduced to a 
decimal by dividing the Ie& or numerator by the denominator, or 
brought to fexagefimals, by multiplying by 6o, &c. Alfo where 
the ubular differences are printed, the fubtradion of the lefs tabular 
number from the greater is faved. 

EXAMPLES. 



z« To find the arc to the natural fine 

•03 1 $474- 
Anf. !*» 48' 28' iz 0315474 

Subtr. I 48, next lefs 03 14 108 

1366 
_6g 

• 2907)81960(28'' 

23820 
23256 

564 
60 



Tab-dift - 



^r^ffm 



2907)33840(12 



l« 



3« To find the arc to logarithm cbfine 

' r9974974- 

6<=> 8' - - 9*9975069 

Anfwcr6° 8' 42" - 9*9974974 

60 
Tab, difierence 136)5700(42" 

ill I 
260 



2. To find the arc to natunl tasg. 

* 1 4326 30. 
Ncztgfeater * • ^43(084 
Anf. 8« 9' 10" 24'" 143263a 
Next lefs, fubt. fr. each 143211^ 



T^b. difitrence 2969)30000(10* 

29090 

1210 
60 



MW 



7a6oo(;j4'» 
5938^- 



««Sk 



13220 

4« To find the arc to logarithm cot; 

10^0905447. 
39'* 4' - . 10*0905978 
Anf. 39^4* 1 2" 20'" 10*0905447 



!)3i86o( 
2581 

k>5< 
16: 



6050 
5162 



888 
60 

, ^58i)5328o(2ow 
5162 

1660 



The 



152 



DESCRIPTION AND USE 



The above method of proportioning by the firft difFerence alone^ 
can oiilv be true when the other difterences are nothing, or very 
final] ', out other means muft be ufed when they are large, viz. for 
the natural tangents and fecants of very large arcs ; and for the loga* 
rithmic fines, and verfed fines of fmall arcs, alio the log. fecants of 
larg^ arcs^ with the log. tangents and cotangents both of fmall and 
large arcs. When the log. fine, verfed fine, or tangent of a fmall 
arc is given, by means of the table of logarithms find the correfpond«- 
ing natural number, and then the arc anfwering to it in the table of 
natural fines, &c. But when the log. tangent or fecant of a large 
arc is propofed, fubtra£l it from 20, the remainder is the log. co- 
taogent or cofine, which will be the log. tangent or fine of a fmall 
arc which is the complement of that required, which complement 
will be found as in the laft remark, by taking; the correfponding 
natural number, and finding it in the natural tangents or fines; 
then fubtra£tingthat complemental arc from 90^, leaves the required 
large arc anfwering to the propofed log. tangent or fecant. And 
when the natural tangent or fecant of a large arc is propofed, change 
it into the log. tangent or fecant of the fame, by taking the log. of 
the propofed natural number ; then proceed with it as above in the 
laft remark. 



EXAMPLES. 



B» To find the arc to oatural tangent 

{O'OOOOOOO. 

20*0000000 
Given 50*0000000118 log.i 1*6989700 



- - - 8*3010300 
*oi97830 nat tan. of 1^ 8' 

2X70 
60 



2910)1 3020o(4V' 
X164 




Hence from - - 
Take the comp* 

Leaves arc reqaireil 



90^ O' o o 
I 8 44 44 
88 51 15 16 



2. To find the arc to natural fecant 
31*6983333. 

20*0000000 
Given 3i*698f its log. 11*5010365 

•oj 15474 - - 8-4980635 
0314108 nat. fmeof 1° 48' 
1366 

60 

2907)81960(28" 
^814 

23820 
233 56 

564 
60 



33840(12 
2907 



/// 



477* 



90 ■ 000 



Hence from - - 

Take the corop. - - i 48 28 12 

Leaves ace required 88 11 31 48 



TRIGONO- 



( 153 I 



rsdbes 



dbs 



TRIGONOMETRICAL RULES. 

'•TN A rigiit-iined triangle, whofe fides arc a,b,c, and their oppofite 
X angles a^byc; having given any three of thefe, of Which one is a 
fide i to find the reft. 






Put s for the fihe, •' the <*ofine, t the tangent,' and t' the cotangent of an trc or 
angle, to the radius r ; alfo l for a logarithm, and l' its arithmetical complemcnu 
Then 

Ca/e u When three fides a,»,c, tffc If thetogtec beright, orss9o'*; 



Put p=:|* A + B + e or feroip erimctcr. 

►PL T j^ — axp — B 

.Then s.ifrrr*/— — *— — . 

* ^ A XB 

And s' f rrrrv/^ii^ . 

^ AXB 

L.$.§r=S:}(l..';^+L.P-.8-f l'A + l'i). 

ATw^, When a =: b, then 

* C f A*— 'r* 



, A , B 

tnen t. a ss <-*r$ t.^ae— • r ; 

B A / 

c = -r- A, ors -<k^B,or=\/A'-t"B • 

8. « S. ^ ^ 

IfA±iB;wefliallhayel ••l<'^ 
ii.a^apo''— f^, and/^'^'T^*^* 

Ca/s 3. When a fide and its oppofite 
angle are among the terms |>iven. 

Then -^ = —•!•= — ; from which 

equations any term wanted may be 
tound* 

Ca/tz.^ Given two fides a,b, and When an angle, as tf, is 90% and a and 
thcir mcluded angle c. c are gi ^en, the n ^ 

p„..-.^^o 1 J, . A — b b =\/a* — c* = v/a + cx a— T^ 
ruts_9o — ff, andt.^z: — --»xt.8; :* -, ,/ -^r^ . 

' A+B And L. Bi=|(L. A+C-f-L. a— C). 

then a :^9+d; an d ^zts— ^. And 

^ » y^AB 8*^f ^^'''^» When two fides a,b, and an 

•-v i*— — — i*. + A -^BT. angle a oppofite to one of them, are 

^'' g>ven ; if A be lefs than b, then ^ r. 

Or in log arithmg, putting l. <^r; c have each two values ; ocb€rwife» 

2L. a-b, and l. 11 = l. ia + l. 2b only one value. 

+ 2L.S.|r— 20, 



we (hall have L.c =2 1 L. (^4- a, 



X 



II.Iii 



JS4 TRIGONOMETRICAL RULES, 

II. In a fpheric triangle, whofe three fides are a,b,c, and their oppofite 
angles <j, b, e; any three of thefe fix terms being given, to find the 



re 



£ 





Qnlike N as r is ZL or T'oo^ Or, 



_ /• A X S. B XS*f C 



•.fc=^ 



rr 



-|-^*r,ACO B. 



Ctf/f I. Given the three fides a,b,c. 
Calliog2P theperim. orp=|.A-^B4-c< 

Then 8. 1 c=rv/- ^^ ^ '• '^^ 



And s'i f f= rv/^ 



8. A X 8. B 



.PX8. P — C 



6. A X 8* B 

L.B.{r=:}(t.S.P— A-I-L.S.?— B-f-L^A+LV.B). 
»r.s'f = J(L.S.P + L.8.P— C+l's.A+l'8.b). 

And the fame for the other angles. 

Ce/e 2* Given the three angles. 
Put 2/= tf +3 +.€. ^Then 

«.|c=rv /'^^'>7^ . And 

■* ^ 8-tfXS.^ • 

1^.8. JC=f(L.S>-f-L.s'/--f+L's fl-f-L'f.^). 

And the fame for the other fides. 

J^aie* The fign < fignifies greater than, 
imd > lefs ; alio co the diffeience. 



In logarithms, put L. <^=r2L# t. f a co b;. 
and L. a = L. 8« A + I** s. b + a l. «• 

|r— 20;thenL 8.{€=:|u <^a« 

G|/^ 4« Given «, ^, and included fide c« 

Firft, let f : s'c :: t, a : t'/ff, like or un^ 
like« as c is <or > 90°; alfoxrs^co ;«• 
Then $'» : &'«» : : t. c : t. a, like or un« 
like « as a is <or> 90*^. 

Or, lct8'fifl+^:s'f«^::t.fc:t.M^ 
> or <9o° as a+i U> or <i8o*' s 

ands. |<?+3:8.|.tfw^::t.|c:t.N,<9o^5 

thenA=rM±N; andB=M=FN. 

Again, let r : s'c : : t. a : t V Hkc or* 

unlike a as c is <or> 90^; 

and ft = 3co m : 

then 8. m*:s.n,:: %'a : sV, like or unlike 

tf as » is < or > h. 

Cafe 5. Given a, b, and an oppofite 

angle 0. 

if(. s. A : s. a : : 8. B : 3. ^, < or ^ 90**, 

2nd. Let r : s'b :: t. « : t'/», likcor unKke. 

B as a is <; or > 90^ ; 

and t. A : t. B : : s'w : s'//, like or unlike 

A as tf is < or > 90"^ ; 

then c zz m±fty two values alfo. 

3dly. Let r : s'tf : t. 3 : t. m, like or un- 
like B as A is <■ or ^ 90° - 



Ca/e 3* Given a,b, and included an- 
gle c. 

To find an angle a oppofite the fide a, ^ ^ y^ f 

Jet r : t'c :: t. a : t. m, like or unlike a, and s'b : s'^a :: b'm ; s'w, Kkc or unlike 

as c is < or > 90° i alfo n:=b lo m: a as « is < or ^ 90° : 

then 8. N : s. M :: t.r : t. ^, like or un- then c = m±k, two vtilues alfo. 

liker^s M is <or> b. But if a be equal to b, or to its fup- 

Or let s'|. a+b : s' {. a co b :: t'|r: t.M, plcmcnt, or between b and its fupplc- 

which ls > or < 90 °a8 a -|- b it>> or < 1 80°; mcnt; th^n is 6 like to B:alfo<*is:s 

ands.|.A+B;s. AeoB::t'if:t.N,<9o°. '"^-^^ *°^ c=:m±n, as b is like or 

Again let r : sV :: t. a : t, m, like or Ca/e 6. Given a^ b^ and an oppofite 

unlike a as ^^ is < Or > 90''.; and fide a. 



as c IS 
K^r BCO M. 

Then h'u : s'k ::5'a 



; s'c, 



like or 



ift.8. a. :s. A ::8.^ : s. b, ^or>90^ 

2od. 



TRIGONOMETRICAL RULE*. 



IS? 



,and. Let r : »'i :: t. a : t. m, like or un- 
like ^ as A if <or> 90"^; 
and t. tf : t. ^ :: 5. M : s. k, <or> 90**: 
ihcD czzu±,Kf as ii is like or uolike 3. 

jdly. Let r : s'a ; : r. h : I'm^ like or un- 
like ^ as A <or > 90^ ; 
ands^^ :t'a :: s. m : 8. », <or>9o®: 
then r= M d: iVy as a is like or unlike h. 

But if A be equal to b, or to its fupple- 
neoty or between b and its fupplen^cnt ; 
then B is unlike h^ and only the lel^ va- 
lues of K9 »« are poffible. 

Nete^ When two fides a,b, and tb^ir 
oppofite angles a^ b^ are known; the 
third fide c, and its oppofite angle c^ are 
readily found thus: 

s. i a CO 3 ; s. i. a-^-h : : t. i a co b ; t. -J c. 
p, ^.Ac/: » : 8. i. A-l-B ::t,i.«co^;t.ftf. 





IIL hi a right-angled fphertc trian- 
gle, where h is the hypoteoufe^ or fide 
oppofite the right angle« b» p . the other 
two fidesi and h^ p their oppofite an- 
gles ; any two of tnefe five terms being 
given, to find che red : the caieS| with 
dieir fblutionsy are as ia the following 
table. 

The fame table will alfo fenre for the 
quadraptal triangle, or that wiiich hat 
one fide = 90% H being the angle op- 
pofite thi»t fidcy B9, p the other cwo an^ 
gles^ and h^ p their oppofite fides : ob« 
ferviog, inllead of h to take its fupple- 
ment; aod mutually change the terms 
liJie and unlikt for each other where m 
is concerned. 



Cafe 



««mMr 



6 

y . w *« 



Given 



9 

B 



R 

b 



.B 
b 



B 



B 
P 



{ 



R^ 



b 
P 



B 
P 

P 



H 

P 

P 



H 
b 

P 



H 



1-. 



H 

B 
P 



S 



U 






%• 



S.H 



s'b 



r 
r 



s.^ 



s'b 



r 
r 
r 



r 
r 
r 



r 
s,b 



r :; s.b : s,^,and is hke b 

t'H ;: t.B : s'^, I ^Qj.> 90^ at H is like or unlike bI 
r :: s'h : s'p, J "^ 



S.H ;; 9,b : s,B, like b 

ib :: t.H : t.p J <;or>90*' as u is like or unlike b 
s'h :: t.b : t'/ J 



r :: s B : s.hI 

t.B :: i'b : s.p > 
r :: s'b : s.pj 



I each < or > 90^ ; both values true 



t^B :: »y : t'u^ < or > 90° as b is like or unlike/ 
s'b :: s./ : %'b , like b 

like/ 



s.b : : t.^ : t. p, 



s'b :: s'f : s'h, < or > 90"^ as b is like or unlike p 

s.p :: t'a : t'^, like b 

s.b :: t^p ; t'p , likep 



t'd r.t'f : s'h, < or > 90° as b is like or unlike / 
r :: s^^ : s'b, like b 
r :: s> : s'p. like/ 



X« 



Th* 



,S6 TRIGONOMETRICAL RULES, 

The following Propofitions and RemariL^, concerning Spherical 
Triangles, (feleSed and communicated by the Reverend Nevil 
Mafkelyne, d.d. Aftronomer Rojal, and FfR.s.)9 will alfo 
render the Calculation of them perfpicuous, and free from Am* 
biguity. 

<* I. A fpberical triangle it equir at its pole; and each angle of the 

lateral, ifofcelar, or fcalene, accord* fametnaogle, will be the Aipplemene 

ing as it has its three angles all equal, of the fide oppoflte to \i in the trian? 

or two of them equal, or all three gle a a c* 

unequal ; and vice verfa, 6. In any tri- 

a. The greatefi fide is always op* angle abc» or 

jsoijte the greateft angle, and the a^c, right anr 

imallefi fide oppofite the' fmalleft gled in A, xft, 

apgle* The angles at the 

3. Any two (ides taken together, hypotenufeareal- 

are greater than the third. ways of the fame 

4* If the three angles are all acute, kind as their op- 

or all right, or all obtuie ; the three polite ddes ; adly, The hypotenufe if 

fides wiU be, accordingly, all lefs than lefs or greater than a quauirant, ac- 

90^, or equal to9o^, orgceaterthan' cording as the fides including the 

90^; and vic€ ver/a* right angle, are of the fame or diffct 

5, If from y'f^ ^^°' ^^^^ * ^^^^ ^* ^^ ^^^* according 

the three an- y^x --\ ^ ^^^^^ ^*™* ^^* *'® cither both 

gles A, B, c, X y^ \ acute or both obtufe, or as one is 

pf a triangle / x \ \ wutc and the other obnife. And, 




ABc,aspoles9 r^^/L \ \ n/ice vrr/a^ ift, The fides including 

there be de- Aj 7<^ ^"^ tA ^^^ '*S^^ aog^^f are always of th« 

icribed, upon ^^.-..i^^ ^J,-^ f*™« J^md as their oppofite angles 1 

the furrace adly, The fides including the right 

of the fphere, three arches of a gre^ angle will be of the fame or difierent 

circle de, bp, pe, forming by their kinds, according as the l^ypotenufe if 

interfedinns a new fpberical triangle Jefs or more than 90^ ; but one at 

pEp ; each fide of the new triangle, lead of them will be of 90?, if thQ 

iHlibethe fupplement of the angle hypotenufe is fu/' 



( «57 ) 



THE CASES OF PI^ANE TRIANGLES RESOlVED BY 

LOGARITHMS. 



IN this and tbe following folutiont 
of fpherical triangles, it is to be 
obferTcd, that when we fay the fine, 
cangent, &c. we mean the logarithmic 
iine, tangent^ &c, as found by the 




Prop. I. Having tht angles^ and one 
fide ; to find either of the other fides. 

Add the logarithm of the given fide 
ko the fine of the angle oppofite to the 
iide required, and ^om the Aim fub- 
tx2£i the {\nt^ of the angle oppofed to 
the given fide; the remainder will be 
the logarithm of the iide rcq )ired. 

kxamle. In the triao|;le bcd, ,har 
▼ingthe angle cdb 90°, cbd ji® 56', 
-B^^ 38° 4', and the fide bd i97'3 j tp 
^nd the fide en. 

2-3951271 log. of i97«3 
9*8961369 fin. of 5 1° 56' 

^2*1912640 fum 
97899.8H0 fin* of 38 4 

2'40i276o log. 25f9278 co rcq, 

Or you may add the complement of 
the fine of the angle oppofed to the 
given fi^ to the two other logarithmi; 
the fum, (abating radius) is the log- 
arithm of the fide required ; as fliewa 
in art. 3, of Log. Arith. And it is to 
be pbferved^ that the complements of 
the fines in the table are to be found 
in the columns of the cofecants : for 
(pafling over ^e firil unit), the cofci* 



cants of the fame ares, are the com*' 
plements of the fame fines. Alfe the 
complements of the tangents^ are cbp 
cotangentf. 

Examine. The fine of iV^ 4' beinf 

97899880, the cofecant of 38^ 4' h 

I0|2iooi2o, which (omittinj^ the firil 

unit), is the complement orthe faid 

fine. 

p-2 looi ;?o CQ, of fin. 38® if 
2*295^1 271 log- of 197-3 
9*8961369 fin. of 51^' 56' 

2-4012760 log. 251*9278, as be& 

But if one fide and the anglea of a 
right-angled triangle be known, and 
you would have the other. fide, as ta 
the former example, the operatiott 
will be eafier, thus ; 

Add the tangent of the angle oppo* 
fite to the fide required, to the logar- 
ithm of the ^iyen fide, the fum (abs- 
ting radius) is the logarithm of the fida 
•required* 

10-1061489 tang. 51^ 56' 
2*2951271 log. ot 197-3 

2-4012760 log. 251*9278 as bef« 

Prop. IL Halving two fides^ end em 
angle oppofite to one of them \ to find 
the otbtr t^o angles^ and the third 
fide* 

Add the fine of the angle given, to 
the logarithm of the fide adjoioiaff 
that angle^ and from the fum lubtxaft 
the loganthm of the fide oppofite to 
that angle, or add its ar* com. the 
remainder or fum will be the fine of 
the angle oppofite to the adjoining 
fide. 

ExamtU* In the triangle aic. 
having tne fide ac 8oO| bg 320, and 



ijS OF RIGHT-LINED TRIANGLES, 

die angle abc 128^ 4^; to find the Tlie fum of the gi^en fides is 882, 

as^Iec BACf ACBy and the fide ab. and the difierence 242, the half fum of 

.the unknown angles is 25^ 58'* 

7-o^69ioo an com. log. 800. 7-05453 14 pomp, log- 88 j 

^•5051500 log- of 320. 2-3838154 log, of 242 

9*8961369 fin. 1 38<^ 4^ 9-6875402 tang. 25^ j^" 

9*4981969 fin. 18 21 BAC. 9-1258870 tang- 7 37 

HaTing BAC and ABC, the angk Angle acb - ^ITTs fum, 

ACB IS their complement to iSoJm. ^^1,^ cab - . i8 21 dif, 

««o ic'i and you may find the fide AS ^ *_ ^ ... .. j ^ a 

by thVfirft pfbpofition. Ji^^fe 7^ 37' being added to 25^ 

vy i.«w tr r- jg/ ^Y^^ jj^|£ jygj ^f jhe angles un- 
known, the fum if 33^ 35^ ^^ the 

l^fop. III. ir^ini tW0 /Jesj and neater angle acb', and the fame 7* 

the angU hetwen them ; to find the ^^ y^^^ fubtraacd from 25° 5S', the 

^tber two an^ks^ and the third fide, remainder is 1 8° 2 1 ' for the leffcr an<» 

glc cab. Laftly, knowing the anv 

If the angle included be a right gies, and two fides, the third-fide may 

Angle, add the radius to the logarithm 'i^^ found by the firft propofition* 
of the lefler fide, and from the fum 

fubtraa the logarithm of the greater p^p^ jy. Having the three fides | /# 

(ide, or add its ar, com. the remainder y x^^ ^ny angle. 
or fum wUl be the taii|;ent of the angle 

Oi^ofed to the leiier &9. Add the three fides together, and 

BxampU. In the triangle Ben, ha- take half the fum, and the differences 

^inj; thefidesE 197*3, and cd 251*9; betwixt the half fum and each fide : 

to mid the angles bcd, gbd, and the then add the complements of the log« 

ftfe cb. arithffis of the half-fumi and of the 

rS9877*8 ar. com. tog. J5i-9 f* M^^p^K the JngkZg^rt 

iTfaiiiZi rad. + log. 197-3 ^^ ^^ Warithms of the dileiencel of 

9-8938989 tan. 38<> 4 bcd. ^^ half fum and riie oth^r fides ; half 

But if the angle included be bb^ their fum will be the tangent of half 

Hque ; add the logarithm of the dif- the angle required. 

fcrenceof the wen fides to the tan, ExampU. In tl^ triangle abCj 

gent of half the fum of the unknown having the fide ab 562, ac 800, an<t 

angles, and from the fum fubtraa the bc 320, to find the angle abc 

logarithm of the fum of the given fides, ac = 800 — ° - "* " 

or add its. complement ; the remainder ab ^ 362 

or fum will be thetangent of half their '/"' ^^^ 



H = 84i. • < CO. 7'07520Ao 

H — AC = 4X . CO, %'l%llli% 

H — A.BSS179 . i'4456o4.» 
H — BC^Sil • . *7i68377 



di^erence* ■ _ • ■ p j - 

Exampk. In the triangle abc, ha- k^^^H^ ^ « fum 206.48620 

ving the fide AB 562, Bc 320, and Tang-of64*'2*=|fum 10*3134310 

the angle-ABC 128^ 4*^; to find the an- Whofe double 128^4^ is the angi^ 

gles BAC> ACB) and the fide ac* ^bc. 



TH9 




ItHt tASfil bt $)>HVRICAL fftlAMGlftS USSdtVfiD 

SY LOGARITHM*. 

■ 

TIE refblutldn of fjfterlcid h<« Prq>. III. Enviitg ffk h^dtiHufe^ 0U 
angles is to be peifbrmcd by the ^m pf the angles; tofihd tbH otbtr 
table of fines, tingeflils afid fecants; -angU. 

toon, followmg, whereof t6 are of ^ j,,^ ^ of Ac angle gWthe 

nght-angled, .nd 12 are of obUque ^ (.bating radios)^!, Ihe wtah. 
cnuigea; andfirft. gent ofthea^glertquired. 

/!/• .1 • i. - » 7 . 1 Example. In the rig^t^angfcd trU 

0/ the nghhangUd manges. . angle abo, bavkg tbe hypetenufe 

AB 30°, and the angle ABC 09° ja-' J 
jS to find the angle B AC. 

9'937S J66 colin. ibyp. a«. 36''ba' 
io'4a4i896 rang, abc •> 69 i* 
io'36i73ox cotaD« bac » 2^.30 ' 

Prop. IV. Hav!ng the hypttenu/k, Und 
ene of the angles ; to find the kg next 
Prop. I-. Having the kgs ; fvjtni the the given angle t 

bypotenufe. Add the timgent of the hjrpotmure 

Ajj t. f c , to the cofine of the angle given i the 

-JTJ^-!!^, ''"! '5«' *°*''* '■"«'' (abating raditu) ia the tangent 
wfinerfrhe«berleg{thefum,cba. of the leg required. ^ 

OBg radma) u the coEne of the hypo- ExampleAn the right^mgled tri- 

''"p if?,""?^* V • 1. , J . ' "»8>« A»c. hwing the bypotenufe • 
.-M**^^*' Jr •'*•* nght-angled tri- ab 30% and the angle abc (J^t-, t» 
•ngle ABC, hivirt AC *7» 54', and find »he leg bc. 
>c M° 30 J to find AB the hynote- _ _a . « , 

«iire» ^*^ 9*7614393 tang. byp.^B jo'oi/ 

5-99ii9a> cofin. .,« 30' Sl^i^ cofi.. abc - 69 a* 

9-9375*98 cofin; 30 AB req. p^p. y. Having the hptten^e, ^mi 

vk TT rr . > . - one of the angles i to find tbe let Op* 

Frop. 11. Having the two legs j z*/,^ ^^a^ ,, ,^ %,^ ^ ; * '' 

either of the ang.es. ^^^ * ^ 

. .. . ^ . Add the fine of the hypotcnufe to 

Add the fine of the leg next the angle ' the fine of the angle given ; the fum, 

fought, to the cotangent of the other (abating radius) is the line of the leg 

leg; the fum, (abating radius) is the required. 

cotangent (rf'the angle required. Example. In the right-angled tri-, 

Eitample. In the right-iingled*tri- angle abc, having the bypotenufe 

angle ABC, having ac if 54', and ab 30°, and the angle «ac 23° 4©'; 

».c 1 1^ 30 ; to find the angle bac. to find the leg bc. 

9-6701807 fin. next leg 2f 54' 9-6989700 fin. hyp: ab 30^ oo* 

122.9153,74 cot. opp. leg. 1 1 30 9'6oo&997 fin, bac - 23 30 

io-36i7i8i cotao. bac 33 30 9-2996697 fin. bc - zi 3© 

- Prop, 



u< 



Tit E R Ei^S O LU T to N Of 



Prop. VI. Having Mi of the Ugs^ and 
tU angle next it ; tofiul the iypetenu/e* 

Add tbe coungent of the given leg, 
to the cofine of the given angle ; the 
fuui, (abating -radius) it the cotan^*- 
g^t ol the hypotenufe required. 

Example. In the righti-aogled trian- 
gle ABC, having th^ leg ac if j/, 
and the angle bag 23^ 30' } to find tjie 
tjrpocenufe ab« 

10*2761(63 cot. AC - ^1^ W 

9*9623977 cof. BAC - 23 30 
20*2385 540 cot. hyp. AB 30 00 

Flop. VII. Hanfing »ne cf the legs^ 
• mnd the^ angle next it ; to find the 

ather leg* 

Add the fine of the leg gtren to the , 
tangent of the angle given ; the fum, 
(abating radiut) is th^ tangent of the 
leg, required. 

Example* In the right-angled trian- 
gle ABC, having the leg ac 27° 54', 
andtheangleBAC^23^3o'; to find the 
leg BC* 

y670iSo7 fin. ac 27® 54' 
9*6383019 tan. BAC 23 30 

9*3084826 tan. BG zi 30 

Prop. VIII. Having one of the legs^ 
and the angle next it ; to find tU 
; ather angle* 

Add the cofine of the given leg to 
the fine of the given angle; the fum, 
(abating radius) it the cofine of the 

angle required* 

Example. In the right-angled trian- 
gle ABC, having the leg bc xi** 30', 
and the angle abc 69° 22'; to find the 
angle bag. 

9*991 1927 cof. BCIX*'30' 
9*9712084 fin. ABC 69 22 
9*962401 1 coi^ BAG 23 30 

Prop. iX. Having one of the legs, and 
iU angle oppefid unto iti to find the 



e* 



Add the radius to the fine of the 
IpTcn legi aad from the fum fubtrad 



the fine of the given* angle, or add xtl 
cpfecant ; the remainder or .fum is the 
fine of the hypotenufe required. 

Example* In the richt-angled 'tri- 
angle ABCf having the leg bc 11^ 30% 
and the angle Bac ^f 30'; to find the 
hypotenufe ab. 

9*29965 $3 fin. BC 11° 30' 

0*3993003 cof* BAC 23 30 

9-6989556 fim AB 30 teqd. 

Prop. Xi Homing one •f the legs, and 
tbe angle opppfed unto it ; to find the 
other leg. 

Add the tangent of Che gtveik leg, ta 
the cotangent of the given a^gle ; the 
funi) (abating radius) is the fine of the 
leg required. 

Exampli* In the right-angled tri- 
angle ABC, having the leg bc 1 1® 30% 
and the angle bag 23° 30'; to find the 
leg AC. 

9*3084626 tang. BC 11° 30^ 

10*3616981 cot. BAG 23 30 

9*6701607 fin. AC 27 54 

Prop. XI. Having one of the legSf and 
the angle eppofid ante it ; to find the 
other angle* 

Add the radius to the cofine of the 
given angle, and.from the fum fubtrad 
the cofioe of the given leg, or add the 
fecant; the remainder or fum is the 
fine of the angle required. 

Example. In the right-angled tri* 
angle abc, having the leg bc i 1° 30% 
and the angle ba<^ 23^ 30'; to find the 
angle abc. 

9*9623977 cof. BAC 23® 30' 
0*0088073 fee. BC II 30 

9-9712050 fin. ABC 69 12 

* 

Prop. XII. Hawing one offhe legSf and 
the iypolennfei to find the angle next 
the given leg. 

Add the tangent of the given leg, to 
the cotangent of the hypotenufe, the 
fum (abating radius) is the cofine of 

the angle required« 

Mxampk* 



OF SPHBRICAL TRIANGLBS, 



i6i 



Sxsmple. In the right-angled tri- 
wif^le ABC9 having the leg ac 27"^ 
54', Uid the hypotenufc ab 30°; to 
find the angle bac. 



23'' 30', and the angle abc 69^ 21'; 
to find the hypotenufe ab* 



^3^30' 



10 



* a 3^5 6 06 



27" 54' 



cot. AB 



/ 
30 



00 



9*9624042 COf. BAC 2 J 30 

Frop. XIIL Halving cm cf tbt legu 
amdthi hypotenuft ; to find tbt angle 
0pfofid to tbt given Itgp 

, Add the radius to the /ine of the 
giTcn legy and from the fum fubtra£t 
the fine of the hypotenufe, or add its 
Gofecant ; the remainder or fum will be 
the fine of the angle required. 

Example^ In the ^ight-angled rri- 
angle ABC, having the leg bc 1 i"^ 30^ 
and the hypotenufe ab 30^; to findxhe 

^nglCBAC. 

9*2996553 fin* teg BC n° 30' 
0^^010300 cofec. hyp, ab 30 og 

9*6006853 fine of pAC 23 30 

prop* XIV* natifing tnt of tbt Ar^/, and 

the bjfottnu/e i to find tbt otbtr leg^ 

Add the radiys to the cofine of the 
•^potenufey and from the fum fub* 
vnidL the cofine of the given leg, or 
add ita fecant ; the remainder or fum 
is the cofine of the leg required. 

Example* In the right-angled trian- 
gle ABC, having the leg bc ii^ 30' 
und the hypotenufe ab 30°; to find the 
kg Ajp* 

9*9375306 co£n. ab 30** 00' 
p*oc 88073 fecy BC II 30 

9*94^3379 cofi>^« ac 2^ 54 

p|0p« XV* flawing tbe angle f ; to find 
tbt bypottnufi% 

AiU fhe cotangent of one oblique 
angle to the cotangent of the other 
obuque angle ; the fum, (abating ra- 
dius) is the cofine of the hypotenufe 
required. 

Example. In the nght-anglcd tri- 
lipgle i^9C| having the an^le bac 



0*3616981 cot. BAC 

9*9758104 cor. ABC 69 22 
9*9375085 cof. hyp. A B 30 00 

Prop, XVI. Honing tbt angles \ ^ to find 
titber of tbt legs* 

Add the radius to the cofine of ei- 
ther oblioue angle, and from the fum 
fubtra^ the fine of the other oblique 
angle, or add its cofecant ; the remain- 
der or fum will be the cofine of the 
leg oppofite to the angle, whofe cofine 
was taken. 

ExampU* In the right-angled tri« 
angle abc, having the angle bag 
23° 30', and the angle abc 69^ 22^4 
to find the leg bc. 

9*9623977 cofin. bac 23® 30' 

0*02879 16 cofec. ABC 69 2Z 

9^9911893 co^Ut BC II 39 



OfQhliqne Triangles* 




Prop. XVII. Having tbt tbrufidctf to 
find any of tbt angles* 

Add the three fides, and take half 
the fum, and the difference between 
the half fum and the fide oppofite to 
the angle fought* Then add the co- 
fecants, or the complements of the 
fines, of the other fides, to the fines of 
the half fum and of the faid diftr- 
ence ; half the futp of thefe four log* 
arithms is the cofine of half theangU 
required. 

^ Example. In the triangle szp, ha* 
ving the fide zs 40°, ps 70°, and pa 
J8'' 30'i to find the apgle SP5, 



i6t 



THE RESOLUTION OP 



Pz;=38 30 
28=400 



Sum 14^ 30 

|fum 74 15 

zs =40 o 

Diff. 34 1.5 



cofcc, 0*0270 1 42 

cofec. 0*2058505 

fia. } fum 9*9833805 

£d« dif. 97503579 

2) 1 9*96660 3 1 

cof. »5°47' 9*9833015 
zps 31 34 required. 



Prop. XVHI. Having the three angles ; 
to find any i>f the fides* 

Let the angles be chabged into 
£de8, taking the fupplemeut of the 
greater; then the operation wiil be 
the Dune at in the former propofition. 

prop. XIX. Having two attglest and a 
fide eppefed t§ one of them ; to find tie 
• fide ofpojed to the other angle* 

Add the fine of the fide giren to the 
fine of the angle oppofite to the fide 
required, and frotn the fum fubtra^SI 
the fine of the angle oppofite to the 
fide given, or add its cofecant ; the 
itmamder or fum will be the fine of 
the fide required* 

Example* In the triangle szp, ha* 
Ting the angle, szp 130° 3' iz'\ 
8PZ 31*^ 34' 26^^ and the fide zs 40°; 
to find the fide PSf 



fin. PR 



tan. ZPS 



cot. ZSP 



i34 7 ? 



30 
26 



40^ 



a 



9*8080675 fin. ZS 

9*9729842 fin. PS reqd. 70 o o 
oee pj. 168 JFollowing. 

Prop. XX. Having tvoo angles^ and a 
fide oppofed to one of them ; to find the 
fide between the anghs given* 

tet ^ perpendicular fall from the 
angle unknowo, upon its 0}>p »fite fide : 
then add the cofineof the given angle 
Belt the given fide, to the tahjjcnt of 
the given fide ; the fum, (dbating ra- 
dius) is the tangent of the firlt sfrc, 
COmptiSbS ended between the given an- 
|rle next the given fide, and the feg* 
ment of the fide where the perpendi- 
cular falls. 



And the fecond arc comprehended 
between the fame fegment and the 
other angle, is to be found thus : 
add the fine of the arc found, to the 
tangent of the griven angle next th^ 
given tide, and from the fum fub* 
trad the tangept of the other angle 
given, or add its cotangent ; the re- 
mainder or fum will be the fine of th^ 
fec<ndarc. 

The fum or difference oftheie 
two arcs will be the fide required. 

Example* In the triangle szp, 
having the angle zps 31® 34' 26'', 
ZSP 30*? 28' i2", and the fide pz 38** 
30' ; to find the fide sp. 

9'930fl78i)cof.zPs (S^^^JS'-" 
440/ \ . —34 

9*9006052 tan. PZ 38 30 o 

9?83i0273 tan. PR iftarc34 7 jq 

9*7488698 " 

932 
97884529 
1227 
0^2301404 

^313 

977679103 fin. SR 2d arc 35 52*39 
add PR lit arc 34 7 30 

fuip is SP 70 o o 
See page' 16$ following. 

But when the perpendicular fall* 
out of the triangle, the difference of 
the two arcs will be the fide requiredt 

Prop. XXI. Having two angles^ etnd 
a fide oppofite to one of them ; to find 
the third angle* 

Let a perpendicular fall from the 
angle unknown, upon its oppofite 
fide : then add the cofine of the givea 
fide to the tangent of the adjacent 
angle ; the fum, (abating radius) if 
the cotangent of the firft angle to 
be found, comprehended by the giveii 
fide and the perpendicular. 

And the fecond angle, comprehend- 
ed by the perpendicular and the fide 
unknown, is to be found thus : add the 
(Ineof the angle found, to the cofineof 
the given angle oppofite to the givei| 

fide* 



/30 *9 t 
1 • -4» 



SPHERICAL TRIANGLES. i6j 

/ 

fide, and from the fumfubtrafi the co* Now add the fine of the firfi s|rc^ 

line of the other angle given, or add to the tanjient of the given angle, and 

itt fecanr ; the remainder or Aim will from the uim fubtrafl the fine of the 

be the fine of the fecood angle. fecond arc, or add its cofecant ; the re- 

The fum or difference of thefe two ttoainder or fum will be the tangent 

Angles will be the angle required. of the angle required. 

Examfli* In ' the triangle szp. Example, In the triangle 8ZP, 

having the angle zps 31^ 34' z6'\ having the fide pz 38^ 30'^ ps 70°, 

28P 30^ 28' 12^% and the fide Pz 38^ and the angle zps 31° 34^ 26''; 10 find 

30^ ; to find the angle szpi the angle zsp. 

^•8935444 ^Gn- '* JS-JOV' 9-9J037«'?-.fia. -,, J3<°34'.'' 

9-7884S«9 lung. ZPS (3' 3* \ ^44oi*=°**"* "* 1 • • *« 

I237 J * [^ . . ao g'90o6o;x tang, pz 38 30 o 

9'682i20Q cot. I ft < pzR 64 18 jo 9*8310273 tan. PR, I ft arc 34 7 30 

9^95476191 . ;64 »8 . taken from Pi 70 o f 

coy J L • • 5* leav e>SR,adg rc 35 52 30 

9-93539481 „..„, \io .9.' 9' 748 8698 V ^^.^ r 34 7 ".' 

, 594/ \ -4« 9J2/ "■"* \. .30. 

9-9598447 fin. 2d < szR 65 44 21 0-2326011 1 r J 35 S3 • 

then addift < pzr 64 18 50 873 / ^«^*«^*^» \ . ..-30 

the fum is szp 130 3 ii 9*7696270 tan. zps req« 30 28 za 

See page 168 f( llowing. . See page 168 following. 

But when the perpcndKular falls T^ find hcth the unknown anglts. 
out of the triangle, the dirtcrence of 

the two angles will be the angle re- Add together the cofecanr, or the 

quiied. complement of the fmci of half the fun^ 

of the given fides, the fine of half their 

Prop. :{XII. Having two jU-s, and <Jjffcrence, and the cotangent of half 

ihe angle between tbem ; to find ei- ^he angle given j the Turn, (abating 

tber of the other angles. radius) is the tangent ot halt the di^ 

rerence of the angles required* 

Let a perpendicular fall from the ^^^ «'^® together thefecant, or the 

unknown angle, which is not required, complement of the cofine. i>i half the 

upon its oppofite fide : then add the ["™ ^\ the jj-iven fides, the cofine o| 

cofine of the given angle to the tan* ***" ^^^ difierence, and the cotan- 

gent of the given fide oppofite to the %^^^ ?^ ^^^\ *^« angle given ; the fum, 

angle required; the fum, (abating (abating radius) is the tangent of half 

radius) is the tangent of the firft arc, "^tr™ °j f^^ "^^^^ required, 

comprehended between the ^vcn an- ***^'* *^? the half diftrenceof the 

file and the fegment of the given fide angles required, to their half fum, and 

where the perpendicular falls. y^" "^^^ ^a^c the greater angle j and 

And the fecond arc is the difference |"^''J^ '^^ half^liffcrencc Irom the 

of that fide and the firft ai c, being com- balf-fum, and you will have the leffet 

prehendcd between the fame fegment f^%^^ required, the fame u in tho 

sMid the angle required. former operation. 



164 



THE RESOLUTION O*' 



PS = 


^^' 


0' 

4 




PZ = 


3B 


30 




Sum 


108 


30 




Diff. 


31 


30 




i Sum 


54 


'S 




JDiC 


»5 4S 


j6" 


2LZPS ' 


= 3» 


34 


J/.ZPS 


s=:i5 47 


>3 



Cofec. ^ fum 0*0906719 

Sin. idifF. 9-4336746 

Cot. Jzps 10*5486352 



Sec. f Aim 0*23 346 r ^ 

Cofio.|diff. 9'98238o< 

CoT.|zps io'54863 53r 

T.49°47'3o''io'07298i 7|T,8o°i i%2^'io"j6i^iysi 

Half fum of angles required is * 80° 15' 42'. 

Half the difference is « • • # 49 47 3Q 

The greater angle szp is • . • 130 3 12 
I The kfler angle zsp is, as before, • 30 28 12 



Prop. XXIir# Having two/Jes^ and 
tbi angle het^an tbem ; to find the 
third fide^ 

Let a perpendicular fall from either 
of the angles unknown, upon its op- 
pofitc fide : then add the cofine of the 
given angle, to the tangent of th6 fide 
from whofeend the perpendicular is 
let fall ; the fum (abating radius) is the 
tangent of the firft arc, comprehended 
between the given angle and the feg- 
inent of the fide where the perpedicu- 

lar falls. 

And the fecond arc is the difFcrcnce 
of that fide and the firft arc, being 
comprehended between the fame feg- 
ment and the end of the fide required. 

Now add the cofine of the fecond 
. «rc, to the cofine of the fide from 
whofe end the perpendicular falleth, 
and from the fum fubtraft the cofine 
of the firft arc found, or add its fe- 
capt J the remainder or fum will be the 
€ofine of the fide required* 

Example. In the triangle szp, 
kaving the fide pz 38^ 30', ps 70°, 
and the angle zps 31° 34' 26"; to find 
the fide zs. 

9*9006052 tang. PZ . 38 3^ ^ 

9-8310273 tan. PR, I ft arc 34 7 3^ 

taken from PS 70 o o 

leaves s a, 2d arc 35 5 ? 3^ 

9-9085938 ?,^fi„. 3^ S35 53 • 

4573 <- • • 30 

9*8935444 cofin* P2; 38 30 o 

9*8842553 coilin. zs rcq. 40 o o 
See page 168 following* 



Prop. XXIV. Having tvcofides^ and 
the angle oppofite to one of them ; /# 
find the angle oppofed to the »tbet 
fide. 

Add the fine of the angle given^ 
to the fine of the fide oppofite to the 
angle required, and from the fum fub 
tradt the fine of the fide oppofite to the 
angle given, or add its cofecant ; the 
remainder or fum will be the fine of 
the angle requited. 

Example^ In the triangle szp 
having the fide p^ 70°, zs 40% and 
the angle szp 130'^ 3' iz" j to find 



the angle zps. 



/ ff^ 



9-8838|94|r..fup.s„549*S6'- 

9-8080675 fin. zs • 40 o <J 

0-0270142 cofine. PS • 70 o o 

9- 7 1 8996 1 fin, ZPS req.^ 31 34 26 

See pag« 168 following. 

Propr XXV. Having «wo fides^ and 
the angle oppofite to on* of thtm j /# 
find the third fide. 

Let a perpendicular fall from the 
angle between the fides given, upon 
its oppofite fide : then add the cofine 
of the angle given, to the tsingent of 
the given fide next that angle ; the 
fum (abating radius) is the tangent of 
the firft arc, comprehended bciwectt 
the given angle and the fegnient of the 
fide where the | erpeudicular falls. 

Now the 2d arc, comprehended be* 
tween the fame fegment, and the end 
of the fide requirec*, is to be found 
thus: add.tbc cofine of the firft arc, 
to the cofine of the given fide oppo- 
fitc to the angle given, and from the 

funa 



SPftfcRlCAL TkIAK6LKft 



i6i 



fdm fubtnfi the cofine of the other 
frtven fide^ or add tfs fecant ; the re- 
ttiaioder or fum will be the coiine of 
the fecond arc. 

The fum or difference of thefe two 
arcs will be the fide required. 

Examfle* la the triangle «2Py 
having the fide pz 38* 30', sz 40S 
and the angle spz 31^ 34' 26^^; to 
find the fide fs. 

y930378'|cofin.,Pz f^i^JS'* 
440^ ^ . .—34 

9^900605 a tang, pz 38 30 o 

9*8310273 tang, PR, iftarc34 7 30 

9*8842540 cofin. sz • 40 o o 
CM 0645 56 fee. PZ . 38 50 o 

9*9086432 cofin. sR^adarc 35 ja 30 
add PR, ifl arc 34 7 30 

gives PS req. 70 o o 
See page 168 foilo^ing. 

But when the perpendicular falli out 
of the triangle, the difference of the 
two arcs will be the fide required. 

Prop. XXVI. Having t*w0 fides^ and 
t/je an^le ofpofed to one of them j to 
find tSe angle between tbcm. 

Let a perpendicular fall from the 
angle between the fides given, upon 
it's oppofite fide : then add the cofine 
•f the given fide next the given angle, 
to the tangent of that angle ; the I'um 
(abatine radius) is the cotangent of 
the firif angle to be found, compre- 
hended by the giyen fide next the an- 
gle given, and by the perpendicular. 

Now the fecond angle, compre- 
hended by the perpendicular and the 
ttther given fide, is to be found thus : 
add the cofine of the firil angle found, 
to the tangent of the given fide next 
the angle given, and from the fum 
fubtradt the tangent of the other given 
fide, or add its cotangent ; the re- 
mainder or fum will be the cofine of 
Iba fecond angle to be found. 



The fum or the difference of tho 
firft and fecond angles, will be the an« 
gle required. 

Example. In die triangle szp, 
having the fide pz 38^ 30', sz 40% 
and the angle s pz 3 1 ? 34' it"i to fiad 
the angle szp. 

9*8035444 cofin. ift . 38^3o'o'^ 

"'^Cn-S-'" {'.".♦.6 

9-6fe3i2oocotan.PZR, ift/.64 i8 jo 

9*90o6oC2 tang, pz • 38 30 o 
0*0761865 cotan. sz . 40 o Q 

9'6i372i3cofin.8ZR, ad^ 65 44 2% 
add PZR, ift ii 64 i8 so 

gives szp,'req. 1^0 3 i» 
See page t68 following. 

Prop. XXVII. Hanging tw^ angleSp 
and the fide between them \ t9 find 
either of the other fides. 

Let a perpendicular fall from the 
given angle which is next the fide re« 
quired, upon its oppofite fide : then 
add the cofine of the given fide to the 
tangent of the given angle oppofite to 
the fide reauired; the fum (abating 
radius) is the cotangent of the firft 
angle to be found, compreheiided hf 
the given fide and the perpendicular 

And the fecond angle is the diffe* 
rence between the firft and the ffTen 
angle next the required fide, being 
comprehended by the perpendicular 
and that fide. 

Now add the cofine of the firft an- 
gle found to the tangent of the fide 
given, and from the ium fubtrad the 
cofine of the fecond angle, or add its 
fecant ; the remainder or fum will be 
the tangent of the fide required. 

Example. lu the triangle szp« 
having the angle spz Ji* 34' 26^', 
SZP 130^ y 12'' and the fide pz 
38? 30^; to find the fide sz. 



t66 



THE RESOLUTION OF 



f **// 



9*8935444 cofio.rz 28^30^0' 



65 44 2Z 



9«682iioocot* psR, ifl <:64 18 50 
taken from iKP 130 3 la 

leaves sxit, gd « 

9-6368859 1 ^^, p,^ 

9'90o6o5» tang, pz - 38 30 Q 

9'9a3Si^ tan. sz req. 40 o o 

See p0ge 168 fellowicg. 

Tofindhctb the uninpwu JiJis* 

Add tt)gether the cofecaot, or th? 

complement of the fine, of half the fum 

of the angles giveni the fine of half 



their difference, and the tangent of 
half the given £de ; the fum (abating 
radius) is rhe tangent of half the dif- 
ference of the iides required* 

Add alio together the fecant,- or 
the complement of the cofio^ of half 
the fum of the eiven aogles, the co- 
fine of half their di&ronce, and the 
tangent of half the ^ven fide; the 
the fum, (abatine radios) is the tan* 
gent of half the lum of tbe fides re- 
quired. 

Then add half the difference of tht 
ijdes required, to theirjhalf fnm, andyott 
will have the greater fide ; and fubtrafi 
the half difference from the half fum, and 
you will have the Idfler fide required, 
the fame as in the former operation. 



az9 

«PZ 

Sum 
Dif. 



>3^^ i' 



31 



12 



// 



t rK 



161 37 3 
98 %% 46 



um 
Dif. 



80 48 49 
49 H 23 



in 



3» 30 

19 15 



o 

Q 



Co&c. I fum 0*00 jf 6062 
Sin. I diff, 9'8793527 
Tang, f PZ 9-5430936 
Tang of 1 5° 94280525 



Sec. I fum 07968360 
Cofin. I diff 9*8148437 
Taog. |pz 9' 5430936 
Tang.of5J*'io-i547733 



Half fum of the fides required is . • • 5 J 
Half their diference is •.•...! 



Tb« greater fide sp is 70 

L«iier fide sz is, as before, • » . . « 40 



prop* XXVIIL Having two angles 
imdtbe fide ieiweem tfoemi tg Jin4 
afcf third angle* 



Let a perpendicular £s!l from eitfier 
of die angl^ given, upon its oppofite 
fides then add the cofioeofthe fide 
givtm to the tangent of the given an- 
l^e, fiom wbich the perpendieolar ^oes 
not fill! \ the fum, (abating raditis} is 
the cotangent of the firft angle, com- 
prdiended by the given fide and the 
penendicukr. 

And the feeood angle is the Jife- 
itnce between the iirfi and the given 
siM;k riiat the peroendicular fdll fron^ 
being comprehcDded by theperpendi* 
colar and the fide •oppofite 10 the 
other an|^e gives. 

Now add the fine of the iecond 
single to the cofine of that given angle 
firom which the perpendicular did not 
fall| and from the fum fubtrafi the 



fine of the firfi angle found, or add its 
cofecant ; the remainder or fum will 
be the cofioe of the angle reqiiired. 

Example. In the triangle szpt 
having the angle «zp 130^ 3' i2'^9 
spz 31^ 34' ik/\ and the fide pz 38^ 
30^; to find the angle psz. 

9-8935444 cofin. PZ 3 9 30 o 

9'78«4S^9\tang. spz (^^ 34 - 
1227 J ^ 1 . . 26 



9«68^iaooGQCan.Psa,ifl^,64 18 50 
taken from szp 130 3 is 

leaves sza, «d /., .65 44 az 
°'''«'773lcof«;.„» (6419 . 

lOI J \ . — 10 

9-930378. )„fi^.„ 

440J 
9-9S9«f46lfi„^3^^ 



3* 35 

J65 44 • 

. , 1 • • ^2# 

9'(^35455o cofin. pszreq. 30 28 o 

See page 168 following. 

ron 



209 



SPHERICAL TRIANGLES. ^7 

lOR THB USE OF THE VERSED SINES MAY B£ ALSO ADDED 

THE FOLLOWING PROPOSITIONS. 

Prop, 1, Hanging two fides of aj^hf the angle szp 136^ 3^ xz"\ to fiadthe 

ric triangle^ wtb the angle between ilde ps« 
them I to find the third fide* 

Ar\T\ .x. % ; , i* r t "^^^ *°8^^ vzp is the fupplcittenc 

DD the lof . verfed fine of the of szp. 
contained angle, and the log, 

fines of the two fides together; the 9'SSioj9o log. verf. vzp 49*56'48'' 

fum (abating twice the radius) is the 9794H96 log. fin. ps 38 30 Q 

logarithm of a namber to be found, 9'8o8o675 log, fin. zs 40 o 

which added to the natural verfed fine 9-1 542761 log.of the number 1416C14 

« the differtnce of the two given fides, Nat. vcif.fum fides 78° 30' 8oo6tal 

Aefumyill be the natural verfed fine Nat. verf. ps 70° . , 6ofe 

of the third fide fought. ^ - • ^%13^1 

Or when the conuined angle is a* t^. ^ ^ . . ^ ^ . 

bovc 9o^ add the log. verfed fine of • ™* ProP<>A«wn may be very ufefia 

its fupplcmcnt, and The log fines of !L !! u^ t ^/^f »^? ?f P^, « 

the two fides together ; the fum r\ ^^' ^^"^^^ longitudes and lati- 

(abating twice the radius) is the loga- T^^ ^^ ^^T^J" • . ^**^ ^.^^^» ^ 

rith of a number to be found, and ^^"^d whofc declinations and n^ht af- 

fubtraifted from the natural verfed fine ^^^^^'^^ <>' longitudes and latitudes, 

of the fum of the two given fides, the V" a^ ' ^^ confequently the aj- 

rejnaind^r will be the natural verfed "J^^««»<>' common altitude of two ftars, 

fine of the third fide fought, 2[ •«>*"t«« ^^ tbe f«nt "d time 

* between the obfcrvations, or difference 

Examfle u In the triangle szp. °[ ff»™^«h> being taken, ^hc latitude 

having the fide pz 38^ 50', ps 70^ ^^ ^^^ ^^^ "V «^»ly ^ ^^^^^ 

and the angle zps 31° 34' 26" j to find p_^^ tt rr • . ^ 

the fide zs. ^ > ^'^;^V ^'''^'/'^ ^^ ^«4^' «/« 

X , - o * aff JP'^^'^ ^"^'^^g'^i and the fide ietweem 

9-1703625 log. ver.CnczsP 31/34 ^6'' ^^; /4r>^/^ /*,W W/. 

97941496 log. fine of PZ 38 30 o * 

0-9729858 log, fine of PS 70 o o Let the angles be changed into fidcf, 

8*9374979 log, of the numb* 865960 and the fide into an angle ; nhen do aa 

Nat. vcrf.diff: fides 51° 30' 1473598 in the former propofition, andtheit- 

l^ar^ verfv zs 40^ • . ^339558 f^^^ ^lU be the fupplement of the third 

£:camtii 2. In the triangle szp, ^"jj^ ^f if one of the «ven anglea 

^ingt^efide,z,80 30^zs;oSand :jS.^trfu?t';i,r^^ 



v' 



Th« 



l6» - THE lElESOLtJTION OP 

The following remarks and diredions, for rendering the proper^ 
tional part of a* logarithm always additive, and for ufing e+f^ 
r— /, &c, for sor eke, in the foregoing proportions, 20, 21, 
2>3t> 23> ^5> 26> 27, 28, were communicated by the Rev. Nevil 
Mafkelyne, d,d. aftronomer royal, and f.r.s. the fourth cafo 
having been invented bv him many years fmce, and delivered 
to the computors of the Nautical Ephemeris, as precepts necef-^ 
fary in computing the moon's dlftances from the ftars in fom« 
cafes, and the reft he has now added upon this occaiion. 

*• The Tefult of trigonometrical calculations, will be fometime^ 
Inaccurate, owing to the logarithms not being carried to a greater 
number of places in the table, as will fufficienthr appear from the 
logarithmic differences being fmalU This will nappen where the 
anfwer comes out in the conne of a very fmall angle, or the fine of 
ah angle near 90**. The irregularity of the logarithmic difFerence^ 
of the fines, tangents, and cotangents of fmall arcs, and the cofines^ 
tangents, and cotangents of arcs near qq^, will fometimes afFefl 
the accuracy of the refult, tmlefs allowance be made for feconct 
idifFerences occ.-^— In oblique aneled fpherical triangles, put t, t', 
s, c for the tangent, cotangent, fine, and cdfine of the jft arc or 
angle mentioned in the foregoing propofitions, then in the 2d part 
^f the work^ 

la prop. 20, if the fird arc if very fmall, for s ufe c+t 

a I • * • angle IS very fmall, for s ufe c-t' 

a a - - - arc is very (mall, for s ufe c-^| 

fl3 - .- - arc is near 90°, for— c ufc t-^g 

25 - ■• arc is near 90^, for c ufc s — t 

^6 - - - angle is near 90°, for c ufe s+t' 

37 - . • angle is near 90°, for c ufc s+t' 

98 r • • angle u very fmall, for— 8 ufe t^— c 

This at the fame time obviates the neceflity of finding the firftarc t» 
a very , minute exadtnefs, and of allowing for fecond difierences kc^ 
"which otherwife would be neceiTary in taking out the fine or cofine 
if>[ the fame arc in the fecond part of the work. 

Where the foregoing precepts dired to fubtrad a fine or cofine, 
it will be readier in pra£lice to add a cofecant or fecaint ; and where 
they dire£tto fiibtradl a tangent (which is done in prop. 26) it will 
be readier to add a cotangent* This method b^ing ufed if it be 
required to find the logarithmic fines &c, to the exa£lnefs of a fe-i 
cond, and the logarithm is increafing (as in the fines, tangents, and 
'iecants) write down the logarithm for the degree and minute with-? 
but the ieconds ; apd alfo write down the proportional part for the 
Teconds ; but, if the logarithm is decreafing (as in the cofines, 
cotangents, and' cofecants) write down the logarithm for the next 
l^ater minute, and alfo write down the proportional part for the 
^ewplcinent of the fisconds to 69 ; and proceed in like manner 

¥ritl\ 



SPHERICAL TRIANGLES. 



169 



with every logarithmic fine, cofme, &c. ufed in the work ; the Aim 
of all the logarithms (abating one or two radii or tens in the index, 
according as 2 or 3 locrarithmic fines &c are ufed in the part of the 
work in queftion) will be the logarithmic fine, cofine, tangent, or 
cotangent required. 

Ex. I. To find the log. fine of 34*^ 17' 24" 

Here log. fine of 34° 17' - - 97507287 

And as 60 -.24 or as 10:4:: 1853 :- - 741 

9'750ii028 

Ex. 2, To find the log, cof. of 55"* 42' 36" 
Log. cof. of sf 43' - - - 97507287 
6o;24(6o— 36),orio:4::i853j__-_74i 

9-750802^} 



«// 



Ex. 3. In the triangle pls, given 
P 3=20° 30' 48 
PS =85 3 40 

PL =289 20 O 



to find Ls by prop, 23 ; 
SD being pcrp. pl. 




20° 31* 



// 



cof. 



PS 85 



— 12 

3 • 
• 40 



PD 84 43 43 

P L 89 10 o 

^D 4 26 if 



9*9715404 

----- 95 

tan. 11*0624350") cof. found by f 

- - - 9814 ^ truing tang.< 

for 2d difF. — 5 J from fine ^ 

tan. ii*0349653 

cofec. PD 84° 44' /' 

— 17 
cofm. CD 4 27 • 

— 43 
codn LS 20 53 24 



, 8«9349686 

II '0349658 
10*0018374 

• - - 35 
9*9986888 

- - 70 



- - 9*9704709 

Here, to avoid the trouble of 2d differences, the cofine of ps is 
found by fubtratS^ing the tangent of it (already found) from the 
fine, which is eafily found, becaufe the differences are fmall : And, 
for the fame reafon, the fum of the tangent and cofecant of pp, arc 
ufed inftead of its fecant. 

N.B. The perpendicular {hould always be let fall fronri the end 
of the fide, PS or pl, which diffeis mod from 90% over or under/* 



OF THE TRAVERSE TABLE. 



THIS traverfe table, or table of 
difference of latitude and de- 
parture, in page 338 & 339, is fo 
contriyedy as to have rhe whole in one 
view, and is fo plainly titled as to 
want little or no explication. 

The diftances i, 2, 3, &c. at the 
top and bottom, mav be accounted 
J Of 20f 30 &c. and the 10 as ioO| 



if the minutes of latitude and depar- 
ture anfwcring to the couife be m- 
crcaftd in the iame proportion ; fo 
that if the dillance confiOs of two fig- 
nifiGant figures, the difference of la- 
titude, and the departure, is each to 
be taken out at twice; and if of three 
figures, at thrice. 

Z The 



170 



USE OF THE 



The chief deGgn of this table is 
for the ready and exa^l woiking of 
traverfes ; but it may alio be applied 
to the fdution of the feveral cafes of 
plain failing, and to fome other ufes. 

Prop- I. Having the ccurfi and dif' 
iancey to find the difference of lad' 
titude and departure. 

Seek the courfe on the left-hand 
of both pages downwards, if lefs than 
four points, or 45 degrees ; or if 
greater, ob the right hand upwards > 
and even with it in the double column, 
iigned at the top and bottom with the 
di (lance, is found both the difference 
of latitude and the depaiture. 

Example x. A fhip fails s s w | w 
37 miles ; the difference of latitude, 
and the departure are required. 

Find the courfe 2\ points on the 
left-haod £deof each pa^^e, and even 
with it in the double columns iigned 
3, and 7, the two figures of the dif- 
tance, the difference of latitude for 
30 is 25*732, and for 7 is 6*004 ; 
the fum is 31736 for the whole dif- 
ference of latitude ; and the depar- 
ture for 30 18 15*423, and for 7 is 
3*599, the fum is 29*022 for the 
whole departure. 

Dfff, Lat. Dtp, 

• 25*732.. 15-423 

6*004 • 3*599 

31*736 . .19*022 

Example 2. A (hip fnils s I 49° 
148 miles; the difFcrcnceof latitude 
and the departure are required. 

Find the courfe 49 degrees on the 
right-hand fide of each pa<;c, and 
even with it in the double coiumns 
iigned 10, 4, and 8, the difference of 
latitude at 100 miles is 65'6o6, at 40 
is 26*242, and at 8 is 5*248; the fum 
is 97*096 for the whole difference of 
latitudt'. And the departure at 100 
miles is 75*471, at 40 is 30*1881 and 



at 8 is 6*038; the fum is iiz!69y 
for the whole departure. Thus, 




Dijl. 

1 00 

8 



. * . 



• . 



a • a . 



Diff. Lat. 
65*606 . 
26*242 
5*248 



« • . • 



. • 



Deparu 

•75-471 
30*188 

6-058 



148 miles . 97*096 a • • 1 1 1*697 

Prop. II. Having federal courfe t and 
dijlances, ta find the difference of 
latitude y and the departure* 

Make a table in the following 
manner, and put therein each courfe 
and diftauce ; then find the diffe-^ 
rence of latitude and departure to 
each courfe by the preceding, and 
place them in the proper column ; the 
difference of the fum s of the northinga 
and fouthings, is the whole diffe- 
rence of latiiude ; and the difference 
of the fums of the eaftiogs and weft* 
ings, is the whole departure* 

Example. A (hip from the latitude 
of 50° north, fails according to the 
courfes and diftances fee in the tra- 
verfe table; the difference of latitude, 
and the departure are found at tb« 
bottom. 



r 







\^^ ^ w W 


Coarfes 






•1^ K} 00 0»^C 






• 

«^ 



a: 

• 




5 

•*» 

to— 
&> 

• 


■4» 




•4 


OS 

• 




2: 




CM 

• 




^ "O Vrt 0- 

• ^ . • * 

^ .^ QO /I VI 

oa N >wj vj ». 




3" 




»^ 

■ 
M 

• 


n 

• 

N 

n 


• « 

♦ J:* 

X -si -*k 

nC - 


a; 


ft) 

c 

• 




■ • 
M 


KH\0 ^f* 9s. 

• • • A 

>0 vj M W 
(Al U» 00 

4k -^ ^-^ 


c» 





a 



> 



This 



TRAVER.SE TABLE. 



171 



This propoGtion may be applied 
in the furyeying of large tra^s of land, 
as a county^ &c. and was made ufe of 
by Mr. Norwood in meafuring the 
diflance from York to London, as the 
road led him, andobftrrvin^ the feveral 
bearings by his circumferentor, and 
finding by fuch a table his feveral dif- 
ferences of latitude, and departures, 
whereby he obtained the dillance be- 
tween the parallels of London and 
York pretty near the truth, fo long 

ago as the 3'ear 1635 : '^ "^^y ^ ^^^^ 
in his Seaman's Pra^ice. 

Alfo in plotting the funrey of a 
county thus taken, the circuit flatton- 
lines, though conliding of many hun- 
dreds, may be reduced to a ftw for 
the firft clofing, and the like for the 
intermediates of each line firft plott- 
ed, whereby every (lation may per- 
haps be more truly placed ihan by any 
other method: the diOancesJn the 
table may be chains of 66, or loo feet 
as well as miles, or any other meafure 
that the differences of latitude and de- 
parture would be had in. 

Prop. in. Having the difference ef 
latitude^ and the departure ; to find 
the cour/e and diftance^ 

Seek the given difference of latitude 
and departure, taken tog;ether, in their 
columns, or fhe neareft numbers to 
them ; and the courfe is even there- 
with at the fide, and the diilance at 
the top and bottom : but if the given 
difference of latitude and departure 
cannot be found nearly, take f, 4, 
&c. parr, or any equal multiple of 
them that can be found ; then the 
courfe is even with them at the fide, 
and fuch a part of the diflance, as was 
taken of the difference of latitude and 
departure, at the top and bottom. 

Example, %• Given the difference 
of latitude 59 miles s, and the de- 
parture 6S miles w ; the courfe and 
dillance are required. 

In the double column over 9, even 
with 49^ at the right-hand fide« isfound 



together the given difference of latitude 
and departure ; therefore the courfe ia 
49^ s W) and the diflance 90 miles 

Example 2. Given the difference of 
latitudes 30 miles n, and the depar* 
ture 1 8 miles B ; the courfe and dif- 
tance are required. 

Here th« given difference of lati- 
tudes and departure, or any numbers 
near them, ore rot to be fi und toge- 
ther in the table, therefore taking ^ o\ 
the double of each, the ccurfr is found 
to be 31^ N £, and the diflance 35 
miles. 

Ifote, wA table computed to every 
mile in the diflance up to a 100 miles, 
would mote reaoily folve this exam* 
pie. 

Prop. IV. Halving the departure and 
middle latitude \ to find the dijerrnce 
of longitude^ according to the method 
«/?^4^ W. Jones, £^j F.R.s. 

Seek the given departure, or the 
next lefo number in the columns figned 
lat. even with the given middle latitude 
found among the courfef, and at th6 
top and bottom (figned jdilK) is the 
difference of longitude fought ; i*hich 
if not found diredly at once, may be 
taken out at twice or thrice. 

Example i. Being yeflerday noon 
in the latitude of 37° ^7' ^t and this 
day noon in 38*^ 43^ k, and by the 
table the departure is found 70*92t e; 
the difference of longituJe is required* 

In the column ligned lat. under 9, 
even with 38°, the middle latitude, is 
found 7*0921 ; therefore 90 miles if 
the difference of longitude fought. 

Example 2. Being yefle^day noon in' 
latitude 46^ 25^ N, and this dar at nooa 
in 47^ 35' N, fofhat the middle lati- 
tude is 47*^ N, and the departure it 
found 1 1 2*53 miles w ; required the 
difference or longitude ? 

In the column (igned lar. over 10 

at the bottom, even with 47*^ at the 

Z 1 right- 



172 



OF MERCATOR'S 



right-hand fide, is 6*8 2Q0 ; therefore 
fubdufting 68'20o from 112*53, the 
remainder is 44*35 ; then over 6 is 
4*0920, and 40*92 fubdudled from 
44' 3 3 leaves 3*4 1, which is found over 
5 ; wherefore the difFerence of lon- 
gitude is 165 miles wed. 

If the middle latitude be not an 
even degree, but have odd minutes ; 
find the diiFercnce of longitude, for 
the even degrees next lefs and greater, 
and add a proportional part of the 
difFerence between the two refults to 
the leiTer ; the fum will be the diffe- 
rence of longitude fought. 

Suppofe the middle latitude in the 
laft example had been 47 ' 20' n; then 
after finding the difterence of lonsji- 
tude as before for 47°, find it alfo 
for 48°, which is 168 miles ; then 
•| of the difFerence being added to 
the former, gives the difference of lon- 
gitude 166 miles wefl. 

Note. Though tns method is not 
in all cafes near the truth, yet when 
the miles are geographical, it is fufH- 
ciently near f^r daily practice in any 
voyage, as well as eafy, and very 
expeditious. 



Prop. V. Having the latiiuda and 

longitudes of fnvo places, tojiudtb9 

hearing and diji^nce* 

Seek the complement of the middle 
latitude among the degrees, and the 
difference of longitude in minutes 
among the diftances, the departure 
anfwering is found in its proper co- 
lumn ; then with the difference of 
latitudes and departure, find their bear- 
ing or courfe anddiftancc by the third. 

Example, • Let the Lizard be given 
in the latitude of 49° 50' n, and 5® 
21 w longitude, and cape Ortcgal 
in the latitude of 44° 10' n, and 70° 
43' w longitude ; to find the bearing 
and diltance. 

The diir'erencc of longitude is 142'; 
and in the columns iigned dep. under 
10, 4, and 2, even with 4^° the co- 
middle latitude, are found 6-8200, 
2*7280, and 1*3640 ; then increafing 
the two former as before (hewn, their 
fum is 96*844 miles w, for the de- 
parture ; and the bearing, or courfe, 
anfwering to 340 miles difFerence of 
latitude, with 96*844 departure, is 
found about 16"^ s w ; and the di« 
itance about 354 miles* 



^ipi 



OF MERCATOR S SAILING. 

THE ufes of the table of meridional parts are fully fupplied bjr 
the table of logarithmic tangents, as is demonflrated in N*' 219 
of thePhilofophical Tranladions. It is there proved, ift. That the 
meridional line, or fcale of Mercator's Chart, is a fcale of the log. 
tangents of the half complements ot the latitudes* adly. That fuch 
log. tangents of Mr. Briggs's form, are a fcale of the differences of 
longitude, upon the rumb'v^hich makes an angle of 51^ 38' 9" with 
the° meridian, And 3dly. That the difFerences of longitude, on 
different rumbs, are to one another as the tangents of the angles of 
thofe rumbs with the meridian. 

Hence it follows, that the difFerence of the log. tangents of the half 
complements of the latitudes, is to the difFerence of longitude a £hip 
makes in failing on any rumb from the one latitude to the other, as the 
tangent of 51* 38' 9'' (whofe logarithm is i0"ioi5iO4) to the tangent 
of the angle of the rumb or courfe with the meridian ; fo that : 

I. If two latitudes, and the difFerence of longitude be given, the 
courfe and diftancc are readily deter;nined by this rule. 

Take 



SAILING. !>} 

Take, by help of the tables, the difference of the log. tangents of 
the half complements of the latitudes, efteeming the laft three figures 
to be a decimal fradion ; and add the complement of its logarithm to 
the logarithm of -the difference of longitude reduced to minutes, and 
the conftant log. lo 1015104; the fum (abating radius) fliall be the 
log. tangent of the courfe. And to the log, fecant of the courfe, add 
the logarithm of the difference of latitude reduced to minutes, the fum 
(abating radius) fhall be the logarithm of thediftance in minutes. 

Example. Given the Lizard to be in latitude 49^ 55' N, Barbadoes 
in 13*^ 10' N, and their difference of longitude 53° 00% orjiSo'Wi 
to find the courfe and diftance. 

'■Colt /^^b^i^^s 38° 25' 1. tan. 9*S993o82 I. 3i8o'=:3'j02427i 
* uLizard 20 2^ 1. tan. 9*5620477 conll.log. 101015104 

diff. 3372*605 its CO. log. 6*4720346 

Log- tang, of the coarfe 49° 59' 16'' sw • io'075972i 

Log. fee. of the courfe 49 59 10 . lo* 191 8067 

Log. of 2205^ diiF. of the latitudes 9'3434o86 

Log. of 3429*378 diftance of Barbadoes from the Lizard . . 3*5352153 

II. If two latitudes and the courfe be given, the difference of lon- 
gitude is obtained urith the fame eafe : for as the tangent of 51^38' 
9'' is to the tangent of the courfe, fo is the difference of the log. 
tangents of the half complements of the latitudes, to the difference 
of longitude fought. Wherefore to the complement of the conftant 
log. 10 1OJ5104, add the log. of the difference of the log. tangents 
of the half complements of the latitudes, and the log. tangent of the 
courfe, the fum (abating radius) will be the log. of the difference of 
longitude in minutes. 

Example, Given the latitudes 49° 55' and 13° 10', and courfe 
49° 59' 'O '5 to find the difference oflongitude. 
Lat. 13° 10', its i CO lat. 38° 25' !. tan. 9*8993082 

Lat. 49 55 ... 20 2i 1. tan. 9-5620477 CO. conft.Iog.9'8984896 
^ , . ^^^' 3372*605 • • its log. 3-5270654 

Log. tang, of the courfe 49° 59' 10'' 10-0799721 

Log. of 3 1 80'= 53° for diff. of longitude 3-5024271 

By this rule, having two good obfervations of the latitude, and the 
courfe duly fleered, the reckoning of a fliip's way is beft afccrtained, 
efpecially if you fail near the meridian. 

III. If the latitude departed from, the courfe fleered, and diftance 
failed, be given; to find the fliip's latitude, and difference of lon- 
gitude. 

Firft, the latitude is obtained from the confideration that the diftance 
is to the difference of latitude, as radius to the cofine of the courfe 
which is common to plain failing. Therefore to the log. of the diftance 
add the log. cofine of the courfe, the fum (abating radius) is the log, 
of the difference of latitudes ; which difference added to the leffcr la- 
titude, or fubtrafted from the greater, the fum or remainder is the 
prefent latitude : then having the two latitudes, and the courfe, the 
difference of longitude is found by the fecond. 

Example. 



tU MEftCAtOk*S SAILING. 

Example^ Having failed from the Lizard, in lat, 49° 55' Uj on A 
eourfe49° 59' lo'' fouth-wefterly 3429'378 miles ; required what lon- 
gitude and latitude the (hip is found in. 

Log. of 3429*378 the diftancefailed ... 3*^35^153 

Log. cofine of 49° 59' 10" the courfe 9'8o8i9v3 

LogiOf 2205', or 36^ 45' diff. of the latitades • • , • . . V3434086 
Now fubtrading 36° 45' from 49** 55', the remainder 13** 10 n. is the 
latitude the fhip is found in. 

By which latitude, now known, the difference of log. tangents will 
be found 3372*605, and the further procefs in nothing differing from 
the fecond rule, whereby the difference of longitude will be found 

S3" 00'. 

Thus the dead reckoning by the log. line, and daily account of a 
Ihip's way, are duly kept, and the trouble very little more than by plain 
failing. 

Thefe are all the Cafes that occur in practice ; the reft, that are 
moftly fpeculative, are either eafily reducible to thefe, or elfe not to be 
performed by logarithms, and therefore come not at prefent under our 
cognizance. 

But it is to be noted, that both the complements of the latitudes 
arc to beeftimated from the fame pole of the world 5 which may be from 
either ; and therefore if one latitude be n, and the other s, to h^vc 
their complements, you muft add 90^ to one of them, and fubtradt 
the other from 90, and then the operation will be the fame as in the 
preceding cafes. 

Example, Given St. Jago, one of the Cape de Verd iflands, in the 
latitude of 14^56' n; and the ifland St. Helena, in latitude 1$^ 
45^ s, and their difference of longitude 30° 12^ e^ to find the courfe 
and diftance. 

^Co 1 /St. Jago 52<> 28^ . 1. tan. 101144965 1. 1812^ 3*2^8i58x 
*^ lSuHcIena37 7^- . 1. tan. 98790845 conft. log 10-1015104 

23 54* 1 20 i ts CO. log. 6 6281714 
Leg. tang, of the courfe 44^ 1 1 ' 53'' s b . . . ^-98784 00 

Log. fee of the coarfe 44 11 53 10*1445200 

Log. of 1841^ diffli of the latitudes 3'2650538 

Log. of 2567*875 diftance of St. Helena from St. Jago . 3-4095738 

Or if it be thought eafier, when one latitude is n, and the other s, 
you may add 90° to each of them, the fum of the log. tangents of 
their halves (abating twice the radius) will be the fame as the differ- 
ence of the log. tangents of the former. Fpr an example take the 
fame latitudes as in the preceding. 

Thenoo^+I'^''^^-"^^^ S^'^ita half^ 5**" ^^' ^' **°' »o'"4496s 
lhengo.-t\^^ 45=105 4i S I S^ S^i 1. tan. 10-1209155 

The fam (abating twice the radios) equal to the former di (lance 2354* 1 20 

Alfo when both latitudes are of the fame name, that is both N or 
both 9, you may add 90® to each of them, the difference of the log. 
tangents of half thefe fums, will be the fame as of the log. tangents 
of half the complements of thofe latitudes, 

OF 



CIRCULAR ARCS. 



'7f 



CP THE TABLE FOR THE LENGHTS OF CIRCULAR 

ARCS. 

THIS IS table 12, and conftitutes page 340. It contains the 
lengths of every fingle degree up to i8p, and of every minute, 
ifcond, and third, each up to 60. The form of it is obvious, the 
length of each degreed, minute, fecond, or third, immediately folbw- 
ing it on the fame line in the next column* And the two following ex- 
amples will ihew the ufe of the table* 

£;r. I .To find the length of an arc of Exf t. To find the degrees, mi* 

57^ 17' 44'' 48'''. nutcs,. &c. in the arc i, which it 

Take out from their refpe6)iye co- equal to the radius* 

lumos the lengths anfwering to each Subtract from it the next lefs tabu* 

of thefe numbers fingly, and add them lar arc, and from the remainder the 

^U together, thus ; next lefs again, and fo on till nothing 

* remain ; and oppofite the feveral num- 

57* . • . 0*9948377 bcrs fubtradcd, will be the degrees^ 

17' f". • • • 49451 minutes, &c* thus; 

44" * • • • . 2133 

48^" 39 Gtvcn length I'ooooooo 

the fumor i-ooooooo is the S?"* • • • • ^'9948377 
whole ieogtb, and is equal to the ra- , 5 ^^25 

^QS. 17 494? I 

2172 

44'' _g^33 

48'^' ..... ri5 

So that the arc equal to the radioa 
contains 57* 17' 44'' 48'^' 



OF 



»7$ 



COMMON AND HYP. LOGS. 



OF THE TABLE FOR COMPAIRING THE COMMON ANO 

HYPERBOLIC LOGARITMS. 

THIS is table 13, and is the upper part of page 341. It contains 
the hyperbolic logs, anfwering to the firft ico Common logs, and 
is very ufeful for fpcedily changing the one into the other. 

Ex. I. To find the hyp. log. anfwer- Ex. 2 To find the common log. an- 
iog to the common log. 0*95 4242 5. fwcring to the hyp. log. 2- 1972246. 

Subtract continually each next lefs 
tabular hyp. log. from the given num- 
ber, and ftom the remainders; and 
the fcveral common logarithms anfwer* 
ing to thefe tabular hyp. logs, joined 
together, will be the com. log. reqd« 
thus ; 



Beginning at the kft band, and di- 
viding the given n amber into periods 
of two figures each, including the in- 
^kz, take out the hyp. log. to each 
period, omitting 2 figures at the 2d 
period, four at the 3d, and 6 at the 
4tb : then add them all together, chu^ : 



hyp. log. 

2*07 2 3266 

1243396 

5526 

• 5^ 



com. log. 
09 • • 
54 . 
24 

25 * 

9*9542425 2*1972246 anfw. 



09 



given 



54 



^4 



ii 



hyp. log. 
2-1972246 

2"07g3226 

12^8980 

1243396 

5584. 



0*9543425 aniw* 

The remaining pages contain the fmall table of the names and de- 
grees, &c. in the points t)f the compafs, which needs no illuflration ; 
and a copious lift of fuch errors, with their corredlions, as have been 
difcovered in the principal books of logarithms ; among which are many 
that have been detefted by myfelf,both in the Avignon edition of Gar- 
diner, and in Gardiner's own quarto edition, which renders this lift 
more compleat than any former one, and it will be foiind very ufe- 
ful in correcfti|ig thofe books of tables which are alicady in the pof- 
feflion of the public. As to all tbe editions of Sherwin's tables in 
oftavo, the errors in them, amounting to many thoufauds, are fartoq 
fiumerous to be printed in this work. 



TABLB 



TABLE I, 



CONTAINING THB 



LO G A R I T H MS 



OF ALL NUMBERS, 



Fro^i X to 100,000* 



ES 



MMMMWI 

Lo 



iM 1*1 



tmt 



W. 



500 
501 
502 

S£4 

SOS 

506 

507 
J08 
S09 



510 
5H 



rs^ 



09^9700 

6998377 
7007037 

7015680 

7024305 



7032914 
7041505 
7050080 
7058637 
7067178 



516 

517 
518 

520 
521 
522 

5^3 
5*4 



5*S 
526 

527 

528 

529 



530 

53J 

53« 
533 
534 

535 

536 

537 
538 
539 
540 

54« 

542 

543 
544 

545 
546 

547 
54^ 
549 

N. 



7075702 
7084209 
7092700 
01174 
09631 



18072 
26497 

34905 
4329^ 
51674 



60033 

68377 
76705 

85017 

933'3 



7201593 
7209857 
7218106 
7226339 

7^34557 



7242759 
7250945 
7259116 
7267272 

7275413 



7283538 
7291648 

7299743 
7307823 

7315888 



7323938 

7331973 
7339993 
7347998 
7355989 



7363965 
7371926 

7379873 
7387806 

7IQ5723 



Log. 



■ I Ml 



k: 



50 

5' 
52 

53 

54 



55 
56 
57 
58 

59 



60 
61 
62 
63 

64 



66 
67 
68 

69 



70 

7» 
72 

73 

74 



75 
76 

77 
78 

79 



80 
81 
82 

83 
84 



85 

86 

87 
88 

89 



90 

9^ 

92 

93 
94 



95 

96 

97 
598 

599 



•og. 



7403627 
7411516 

74>939« 
7427251 
7435098 



7442930 
7450748 
7458552 
7466342 
74741 1 8 



7481880 
7489629 

7497363 
7505084 

75'279^ 

7520484 
7528164 

753583' 

7543483 
755M23 



7558749 
7566361 

7573960 
7581546 

7589119 



7596678 
7604225 
7611758 
7619278 
7626786 



7634280 
7641761 
7649230 
7656686 
7664128 



7671559 
7678976 
7686381 

7693773 
7701153 



7708520 

77>5875 
7723217 

7730547 
7737864 



7745170 

7752463 

7759743 
776701Z 

7774268 



Log. 



GA R ITH 

Log. 



M 8 



N. 



600 
601 
602 

603 
604 



605 
606 
607 
608 
609 



610 
611 
612 
613 
614 



615 

616 
617 
618 
619 



620 
621 
622 
623 
624 



625 

626 
627 
628 
629 



630 
631 
632 

633 



635 
636 

638 

639 
640 
641 
642 
643 
644 



7781513 

7788745 

7795965 
7803173 

7810369 



7817554 
7824726 

7831887 

7839036 
7846173 



7853298 
786041 2 

7867514 
7874605 
7881684 



7888751 

7895807 
7902852 

7909885 
7916906 



79239*7 
7930916 

7937904 

7944880 

7951846 



7958800 

7965743 
7972675 

7979596 
7986506 



7993405 
8000294 

8007171 

8014037 



634 8020893 



8027737 
8034571 
8041394 
8048207 
8055009 



645 
646 

647 
648 

649 



N. 



8061800 
8068580 
8075350 
8082110 
8088859 



8095597 
8102325 
8109043 
8115750 
8122447 



Log, 



<mm 



w. 



N, coo L,69 



650 
651 
652 

653 

654 



655 
656 

657 
658 

659 

660 
661 

662 
663 
664 



665 

666 
667 
668 
669 



670 
671 
672 

673 
674 



675 
6y6 

677 
678 

679 
i68o 
681 
682 
683 
684 



,og. 



8129134 
8135810 
8142476 
8149132 

8*55777 



8162413 
8169038 

8175654 
8182259 

8188854 



8*95439 
8202015 

8208580 

8215135 

8221681 



8228216 

82H742 
8241258 

8247765 
8254261 



S260748 
8267225 
8273693 
8280151 
8286509 



685 
686 
687 
688 
689 



690 
691 
692 

^93 
694 



695 
696 

6g7 

698 

699 



N. 



8293038 
8299467 
8305887 
8312297 
8318698 



8325089 

8331471 

8337844 
8344207 

8350561 



8356906 
8363241 
8369567 
8375884 
H382192 



8388491 
83947 




■^ 



7908450980 



701 
702 

703 
704 



705 
706 

707 

708 

709 



710 
711 

712 

7^3 

7H 

7^S 
716 

717 
718 

719 



720 

721 
722 

723 
724 



725 
726 

727 

728 

729 



730 

731 
732 

733 
734 



hog* 



8457 i8p 
8463371 
8469553 

8475727 



8481891 
8488047 
8494194 

8500333 
8^06462 



8512583 
85 1 8696 
8524800 
8530895 
8536982 



8543060 
8549130 
8555192 
8561244 
8567289 



8573325 
8579353 
8585372 
8591383 
8597386 



8603380 
8609366 

8615344 
8621314 

8627275 



735 
736 

737 
738 
739 




N. 



8633229 
8639174 
8645 1 1 1 
8651040 
86^6961 



8662873 
8668778 

8674675 
8680564 

8686444 



8692317 
8698182 
8704039 
8709888 
8715729 



8721563 
8727388 
8733206 
8739016 
8744818 



Log, 




I 



M 5 7 'ij. 



N. 1040 L.017 



0170333 

4S°7 

8677 

0181843 

_ 7°°i 



0191163 
5317 
9+67 

DZ03D 1 3 
7755 



'893 



+284 
8406 



3^538669 



656: 



otS^S'S Z9363348l37"*'7' 

6639,7050 7467 7; 
0240750 m6i 15711, 

4857526756786 

896093701978010 
73079134683878;+ 



79728 
Z063 2 
6150:6 



21471915127 



7 « 547563 
12036124511654 
S333S74' 
9416 1 9814 

"65027349613904 ,_ ,. ,. . 
66 7!7»|7979 8387 «794,9»o 
670281644 1051 [245 8;z865;3i7z 

68 57i3.6ii96526693z'7339j 

69 _ 9777oi83 , 059O0996 | i402 

1070 0293838 4244J4649!5=>S5lS46i| 
71 7895|830o87o6;9iii;95i6; 
720301948 i353U758!3i63l3568j 
73 599764^^6807721117616 
74,03100431044710851 1156(1660 



li 



408514489148935296 

8123:852689309333 

032215712560 Z963I3367 

61886590699317396 

' 'P'9|'4^= 



0341273 
, 6285 
350293 



0362295 
6189 

0370279 



50415444 
9060946: 
30753477 
7087,7487 
094 H95 



4698 5098^5498 
I698I9098 9498 
695 13094 3494 
1688 7087 7486 
0678 1076 J475 




1 



■»^ 



wm»> 



<8) 



Logarithms N» 10900 L. 03X 



N; 



1090 

9» 

92 

93 
94 



95 
96 

97 
98 

99 



1 100 
01 

02 

03 
04 



SI 

07 
08 
09 



0374265 
8248 

0382226 
6202 

0390173 



4I4I 
8106 
0402066 
6023 

9977 



0413927 

7873 
0421816 

5755 
9691 



4663 
8646 
2624 



65996996 

05700967 



4538 
8502 
2462 
6419 
0372 



0084 



5062 
9044 
3022 



5460 
9442 

3419 

7393 
1364 



4934 
8898 

2858 



0767 



43224716 
8268 8662 
2210,2604 

61496543 



0477 



0433623401614409 



533» 
9294 

3*54 



68147210 



1162 



5111 

9056 
2998 
6936 
0871 



5858 
9839 
3817 
7791 
1761 



57*7 
9690 

3650 

7605 

1557 



5506 

9451 

3392 

7330 
1264 



755* 
0441476 

5398 
9315 



IlIO 

II 

12 

13 
14 



7944 
1869 

5790 

9707 



8337 
2261 

6181 

0099 



4802 
8729 
2653 

6573 
0490 



5*95 

9122 

3045 
6965 

0882 



04532303621 
7Hi'753> 



0461048 

4952 
8852 



>7 
18 

»9 



1120 
21 

22 

23 
*4 



1438 

534* 
9242 



4012 
7922 
1829 

573* 
9632 



4403 

8313 
2219 



4795 
8704 

2610 



6122 6512 
002 1 041 1 



6257 
0237 
4214 
8iSi8 
2158 



6124 
0086 
4045 
8001 
1952 



5900 
9845 
3786 

77*3 
1657 



6655 

0635 
4612 

8585 
*554 



6520 
0482 
4441 
8396 

*347 



5587 
195 H 
3437 
7357 
1*73 



0472749 
6642 

0480532 
4418 
8301 



3138 
7031 

0921 

4806 

8689 



li 

*7 
28 

29 



0492180 
605,6 

99*9 

0503798 

7663 



N. 



35*8 
7420 

13091 

5195 
9077 



3917 
7809 

1698 

5583 
9465 



4306 
8198 
2087 

597* 
9853 



5186 
9095 
3000 
6902 
0801 



7 



7053 
1033 

5009 

8982 

2951 



6917 
0878 

4837 
8791 

2742 



6295 
0239 
4180 
8117 
2050 



6690 
0633 



8 



7451 
143 » 
5407 

9379 
3348 



73«3 
1274 

5*3* 
9187 

3137 



7849 
1829 

580I 

9776 

3745 



7709 
1670 
5628 
9582 

353* 



7084 
1028 



45744968 



8510 
*444 



5980 
9907 

38*9 

7749 
1664 



8904 
*837 



2568 

6444 
0316 

4184 

8049 



2956 
6831 
0763 

4571 
8436 



0511525 

5384 

9*39 
0523091 

6939 



1911 

5770 
9624 

3476 

73*4 



2297 
6155 
0010 
3861 
7709 



3343 
7218 

1090 

4958 
8822 



3731 
7606 

H77 

5344 
9208 



2683 
6541 

0395 
4246 

8093 



3069 
6926 
0780 

463 J 

8478 



4696 

8587 

6360 
0241 



4119 

7993 
1863 

5731 
9595 



5577 

9485 

3391 
7292 

1190 



6373 
0299 

4222 

8140 

2056 



6766 
0692 
4614 

8532 

*447 



5968 
9876 

3781 
7682 

1580 



DiPro. 



398 



397 



396 



395 



7479 
1422 

5361 

9*97 
3*30 



7«59 
1084 

5006 

8924 

832 



63596750 
0267 0657 



5085 
8976 
2864 

6748 
0629 



3455 

1166 
5016 
8862 



8380 
2250 
6117 
9981 



547458646*53 



9365 
3*53 

7^^ 

1017 



450648945281 



8767 
2637 

6504 

0367 



4171 
8072 
1970 



3641 

75*5 
1405 



394 



398 
1^ 



IIP 

«5P 

199 
119 
»79 
318 

358 



393 



39* 



397 
40 
79 
119 
159 
199 
.38 

t78 
318 

35T 



4561 
8462 

*359 



97540143 



4030 

7913 
'79- 



5669 



9>54954» 
3024 341 1 

689072^7 

07531139 



3841 

7697 

i55» 

5400 

9*47 



4**7 
8083 

1936 

5785 
9631 



4612 
8468 
2321 
6170 
0016 



391 



390 



389 



388 



387 



396 

79 
119 
158 

ip8 
138 

177 
317 
356 



4998 
8854 
2706 

6555 
0400 



8 



386 



38s 



1 



D 



?95 

4<> 

79 
119 

158 
198 

137 

177 
316 

S5<J 



PROPORTIONAL PARTS. 



39a 

3PI 

390 

389 
388 

387 
386 

385 
384 



39 
39 

39 
39 
39 
39 
39 
39 

12 



ilJ. 

78 

78 

78 

78 

78 

77 

77 

77 

II 



118 

117 

117 

X17 
116 
116 
116 
116 

115 



157 
156 

i5<$ 
156 

'55 
155 

«54 
154 
154 



I9tf 
196 

»95 
«95 
194 
194 
»95 
193 
19& 



»3S 
*35 

*34 

*33 
133 

»3» 

»3i 

130 



2. 

*74 
174 

»73 
a 72 
»7» 



270 



J8_ 

3 '4 

1«3 

31a 

311 

310 



353 
151 

351 

^50 

349 
348 
347 
347 
1^91 357134^ 



^1\ 310 



309 



270 308 



383 
381 

381 

380 

379 
378 

377 
376 

375 



I 



38 

38 

38 

38 

38 

38 

38.- 

3875 

38175 



77 
7<J 
7< 
76 

7tf 
76 

75 



"5 
"5 

I«4 
114 

"4 
"3 
"3 
«I3 
"3 



»53 
t53 
151 

152 

'5» 
'5» 
151 
ISO 

12 



i 

191 

191 

191 
190 
190 
189 
189 
188 
188 



»3o 

st9 

219 

«»7 

iii 



1^8 

167 
»66 

165 
264 
2<3 

»«3 



306 
\o6 

305 
304 

303 

302 
301 

300 



-2- 
34 1 
344 
343 

34* 

34* 
340 

339 
338 
338 



394 

S9 

79 

118 

158 

236 
176 

3'5 

355 






393 



39 

79 

118 

»57 

«97 
^36 

»75 
314 

354 



ra 



»mm 



N. xr3QO L. 053 



o F 



N. 



1130 

32 

33 
34 



0530784 
4626 
8464 

0542299 
6131 






35 
36 

37 
3» 
39 



1 1 40 



4a 
43 

44 



6514 



9959 

0553785 
7605 

0561423 
■5237 



1169 
5010 



^553 

5394 
884^9232 



0341 
4166 

7987 
1804 

5619 



3066 
6896 



N U M B E R S. 



1937-2321 

57781616? 

9615,9999 

3449i3«3^' 
7279I7662 



9049 
410572856 
6661 
0580462 
4260 

~8o7^ 
05^1 846 

5634 

, 9419 
0603 200 



9429 



0724 1106 1489 

454849305312 

8369 

21.86 2567 

6ooo'638i 

98io|oi9i!o572 



8750^9132 
2949 
6762 



46 

47 
48 

49 



3237361813998,4379 
T041 7422178028182 
0842' 1 222! 1 602 1 98*2 

4640 15019 :5399^778 



1150 

5« 

52 
53 



6978 
0610753 

4525 
8293 



540622058 2434I281 1 



57 

_i21 



u6o 
61 



843418813,9193,9572 

2225 i?6o4J2983l3 362 
6013 6391 I6770J7 148 
9797 ;o«75 055410932 



2706 



8 



3858 



30903474 
65466929J7313 
0382 0766. 1149 
14215 4598I4981 
8045 8428J8811 9193 

1871 22541S636 3019 
5694 6077I6459 6841 



7^97 
J 53 2 
5365 



4242 
8081 
1916 

574^i383 
9576 



D 



(9.>^ 
Pro. 



3^4 



95149896 
333037*2 
7H37524 



027,8 

4093 
7905 



J09S3M34 
'4759 5 HO 5520 
1856289429322 
[2362,274113121 
6158:653716917 



0659 

4475 
8287 



3401 

7223 
1041 

4856 

8668 



17142095 



5900 



3501 
7296 



2476 
6281 



9702*0082 



9951 
13741 

17527 



03300709 1088 
411 9J449S 4877 
7905182848662 



3881 
7676 



1 3 loi 1 688 |2o66;2444 



1467 
5256 
9041 

2822 



382 



381 



3S0 



379 



374 

37 

75 
112 

150 
187 
XZ4 
261 
299 
337 



t, 

3 

4 

5 

6 

7 
8 



3578i395^4334!47t2J 5090:5468 584516223^601 



7356,7734 81 1 1**5489 18866:9244 962 1 19999 

Ii3i[i508ti885|2262 12639I3017133943771 
'-'^-'•— ^5^5j6o32[ 640916786:716317540 



4902,5279 

867019046 



5820 

9578 

0633334 

7086 

0640834 



45804954 



8322 



620652061 

63 5797 

64 953c 



65 
66 

6S 
69 



1170 

7« 

72 
73 

74 






. 75 
76 

77 
78 

N. 



6196 

9954 

3709 
7461 

1209 



94239799 
3 1 87 13 563 



0330:0705 



8696 

2435 
6171 

9903 



^57216948(7324 



101760552,09291305 
i393_9|43« 614692 5068 



0376 
4148 
7916 
1682 

5444 



4084 
7836 
1584 



5329 

9070 



4460 
8211 
1958 



5703 
9444 



1081 

4835 
8585 

2333 



2809 3182 
65446917 



0276 



0663259 
6986 

0670709 
4428 

8145 
0681859 

5569 
9276 



5940 

9647 

06929803350 

668 1 705 1 



0700379 

4073 
7765 

0711453 
l5«38 



3632 

7358 
fo8i 

4800 
8517 



0649 



4005 

7730 

H53 
5172 

8888 



22302601 
6311 

0017 



3721 
7421 



0748 
4442 
8134 
1822 
5506 



1118 
4812 
8503 
2190 

5875 



2 



6077 
9818 

3556 
7291 

1022 



43774750 
810384751 



378 



377 



376 



373 

37 

75 
III 

149 
187 
224 
7 261 
298 

33<5 



i7699'8o75!845 1 18827 9203 
1145611832*220712583 2958 
J52io!5585'596o6335 6711 
896019335197100085 0460!^ 



2708:308213457 



6451 
0192 

3930 
7664 



6826 
0566 

4303 
8037 



1825 
5544 



2197 
59'5 



9259I9631 



2972 
6681 
0388 
4091 

7791 



3343 

7052 

0758 
4461 

8160 



1487 
5181 

8871 

2559 
6243 

J- 



1857 

5550 
9240 

2927 

66ii 

4 
B 



1395J768 



7200 
0940 

4677 
8410 

2141 



3831 



5>23|5495 5868 
8847192209592 
2569 294IJ33I3 
6287 6659I7030 



7574 

'3>4 
5050 

8784 

2514 



42c* 

794« 
1688 

5424 

9157 
2886 



0002 



0374 



37H 

7423 
1129 

4831 
«530 



2^26 

5919 
9609 

3296 
6979 



4085 

7794 
1499 

5201 

8900 



0745 



6241 
9964 

3685 
7402 

1116 



4456 
8164 
1869 

5571 

9270 



2596 
6288 

9978 
3664 

7348 



4827 

8535 
2240 

5941 
9639 



6613 
0336 

4057 

7774 
1487 



2965 
6658 

0347 



3335 
7027 

0715 



5198 
8906 
2610 
6311 

0009 



374 



373 



372 



372 

37 

74 

112 

149 

186 

260 

298 

?3S 



37* 

37 

74 

III 

148 
186 

»i3 
i6o 
297 

334 



40334401 
7716 8084 

8 



3704 
7396 

1084 

4770 
8452 



371 



370 



369 



D 



370 
37 
74 
III 

148 

i8s 
222 

296 
333 

369 
37 
74 
III 
148 
'85 
221 
»j8 

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72447410 



9077 



07440911 



2184 



7403 
9067 

0729 

2391 

4053 



05260692 



2350 



0383 

*034 
3684 

5333 
6982 



8630 



8795 



02780442 
1925 2089 

35713736 
5217 5381 



6862 
8506 
0150 

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3436 



7 



0903 
2572 
4241. 
5909 

7577 



6076 6243 
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92449411 



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1957 
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7191 

8835 
0479 

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3764 



1070 

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433'9 



1237 
2966 

4575 



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167 



4505 



599^6161 



0713 
2364 
401414179 



89609125 



0772 



22542419 



5570 
7211 

8852 

0492 

2131 



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9164 
0807 
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4093 



57341 

7375 
9016 

0656 

2Z9 



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6i 

63 

N 
6s 

66 



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69 



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7« 
72 

73 
74 



77 
78 
79 



2623 

97I4261 

J899 

7536 

9«73 



8816 



4250449 
Z081 
3712 



7700I7864 
9336 95CX) 



0809 

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0972 
2608 



4079 4?+^ 



5713 
7347 



6972 
8601 



67 4269;J30 



1858 
3486 



5113 



8980 

0012 
^244 

3875 
5505 



5*75 



6739J6901 
^3658527 



9990 
4271614 



0152 
1776 



3400 



Logarithms 
a I 3 Z < i 6 



2786 

4425 
6063 



2950 
4589 



3"4 

4753 



6226 6390 
8027 



5877 
7510 



2771 
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9664 



11361300 



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4569 
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9827 



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9633 9796i9959|oi 22^0286 
U65|i428!i59iji754|i9i7 

2896;3059;3222|3385!3549 
4527I4690 4853:50 j6!5 179 



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87648927,909019253 
0393 0556 

2021 2184 
36483811 



61576320 



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23471*509 
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869o'8852 90i5 

O3i5;o477!o639 

193912101*2264 

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50231518653485510 

6646,6808:697017133 



2680 
81 
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83 
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4861 

6484 

81061826818430*859218754 

972 7; 9889t005i| o2i3:o37(^ 

1672I18341996 



4281348:1510 



2968 

4588 
6207 

7825 



3130 
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7987 



85 94439605 
864291060 1222 

*^- 26772838 

4293'4454 



87 
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89I 



32921345413616 
49i2|5073i5235 



6369 6530.0692 6854 



5908 6070 623 f 6393:65 54 



2690 

9* 
92 
93 
94 

96 

97 
98 

99 



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9137 
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9298 
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39764137 



8149 : 83118472 
9766J9928I0090 

»383i545|'707 
3000316213323 

4616477714939 



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4298 



8007 8169 



94609621I9782 
1396 



5749. 

7360! 

8970 
0580 
2029) 2190 

O 1 I I 2 



5910 

521 

9132 

0741 

2351 



1235 

2847 

4460 



6071 



7682 

9293 
(^02 

2512 



3009 
4621 



3278 
4916 



3442(3606 
5080I5244 



N> 21^500 L> 4331 
8 9 (IT pro. 



835585188682 
999' !oi 54*03 18 



1627 

3262 

4896J5060 

653066936857 

8i63;8327i849o; 



1790! '954 
342513589 

5223i 



2117 

5386 

7020 

8653 



6483i6646!68o9 



7787179508113 
9416195799742 
10441120711370 
2672J2835 2998 
4299.4462^4625 



8276i8439 
990410067 

'533!'695 
316013323 

478714950 



5926|6o88|625 1 164 1 465 76 
7552i77'47877;8o39J82o:? 
9'77 9S4o.9502'9665J9827 
0802 0964I 1 1 27! 1 289' 1452 
2426 2588^275 1 '291 3I3076 



4050,42121437414536,4699 



5672 

7295 
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0538 



2158 

3778 



61596321 
7781I7944 

94039565 
07000862I1024I1 186 



583515997 

745717619 
9079:9241 



163 



2320,2482 2644{28o6'x62 
4264:4426 



3940,4102 . 
5397 5559:572115883 6045 

7016,71787340,750117663 
86348796 8958;9i 19:9281 



025210413057 5:073710898 
i868!203o'2i92|2353!25]5 

3485l3646;38o8;3969'4i3' 
|5iooj5262:5423 558515747 

'67'5i6877i7038j7200j736i 



i8330,849ii8653;88i4;8976 
9944ioi05;o267;0428 0589 
'557l'7^ 81 1 8802041 U202 

3'7o:333ii349*'3653 38iS 
4782.4943:5 104:5265.5427 



6232 

78431 

94541 
1063 

2672 



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130 

H7 



162 



4 
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1301 
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18004,8165.832618487 8648 *^" 
'9615:977699371009810258 
1224-1385:154611707,1868 

2833i2994!3'55i33i6'3477 



5 1 6 i 7 rT"i 9 'DRts>t i 

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i6i 
16 

3» 

48 
64 
Si 

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lip 

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N. 27000 L, 431 



OF Numbers. 



2700 
01 

oz 

03 



2710 
II 

12 

Ji 

18 
19 



4313638 

5246 

6853 

8460 

044320067 

OS 

06 

07 
08 

09 



3798 

5407 
7014 

8621 

0227 



1673 
3278 

4883 
6487 

8090 



9693 

433 "95 

2897 

4498 
6098 



1833 
3438 

5043 
6647 

8250 



3959 

55^7 

7^75 
8782 

0^88 



4120 
5728 

7336 
8942 

0549 



9853 

H55 

3057 

4658 
6258 



7858 



7698 

9298 

4540896 1056 

2495 2<^54 
4092*4252 



2720 
21 
22 

*3 



5689 5849 



7285 
8881 



7445 
9041 



4350476 0636 
24I 207112230 



1994 

3599 
5203 

6807 

8411 



0013 
1616 
3217 
4818 
6418 



4281 
5889 
7496 

0709 



2154 

3759 

5364 
6968 



8018 
94589617 
1216 
2814 
4412 



6008 
7605 
9200 

0795 
2390 



0174 
1776 

3377 
4978 

6578 



8178 

9777 
1376 
2974 

457> 



23x5 
3920 

5524 
7128 



85718731 



0334 
1936 

3537 
5138 

6738 



8338 

9937 
1536 

3'34 
473 » 



6168 

7764 
9360 

09S5 
^549 



6328 

7924 

95^9 
1 1 14; 

2709I 



11 

27 
28 



3665 38243984:4143 
5259;54i8 
685117011 
84449603 



5577157365896 
7170:73297488 

8762(8921 



29 ^436oo35 ; oi94! o354 | o5i3 



4303 



9080 
0672 



2730 

31 
32 

33 

34 



1626:1786 194^ 2104)2263 

32i7'3376 3535:369413853 
48o34966 5i25!5284J5443^ 



6396 6555'67i4'6873 7032 
7985 8144830384628620 



35| 95739732989iioo5o 
364371 161I1320JI478.1637 

2748J2907!3o65t3224 



37 
38 

39 



2740 

41 
42 

43 
44 



46 

47 



0208 
1796 

3383 



4334'4493!46s2;48io|4969 
5920 6079I62 3716396165 5 5 



7506.76647823 798118140 
909019249,9407 9566.9724 
4380675083310991 ii5o|i3o« 
2258|24i6,2575t2733:289i 
384i ; 3999 | 4i5H , 43i6 | 4474 

6056 

7638 
9219 



5423 5582J574O15898 



70057163 



8587 



484390167 
491 «747 



1225 



8745 
03250483 



7322 
8903 



2063 



7480 
9061 
0641 
2221 



4442 
6050 

7^57 
9264 

0870 



2475 
4080 

5685 

7288 

8892 



0494 
2096 

3697 
5298 

6898 



8498 
0097 
1696 

3293 
4891 



6487 
8083 
9679 
1274 
2868 



A462 
6055 
7648 
9240 
0831 



2422 
4012 
5602 
7191 
8779 



0367 

^955 

354^ 
5127 

6713 



0799 

2379I 



8298 

9883 

1466 
3050 
4632 



6214 
7796 

9377 
0957 
2537 



4603 
6210 
7818 
9424 
1030 



2636 
4241 

5845 

7449 
9052 



0654 

2256 

3858 

5458 
7058 



8658 
0257 
1855 

3453 
5050 



6647 
8243 

9838 

H33 
3028 



4621 
6214 
7807 

9399 
0990 



2581 
4171 
5761 
73SO 
8938 



2113 

3700 
5286 
6872 



8457 
0041 

1625 

3208 

479 > 



4763 
6371 

7978 

9585 
1191 



2796 
4401 
6005 
7609 
9212 



0815 
2416 
4018 
5618 
7218 



8818 
0417 
2015 

3613 
5210 



8403 

9998 

^593 
3187 



8 



4924 
6532 
8139 



9746 9906 
1352 1512 



2957 
4562 

6166 

7769 

9372 



0975 

2577 
4178 

5778 
7378 



8978 

0577 
2175 

3773 
5370 



6Sof6g66 



8562 
0157 
1752 
3346 



1: 



47814940 
63740533 



7966 

9558 
1149 



8125 

9717 
1308 



2740 2899 
43304489 
59206078 
75097667 
90970256 



0526 0685 



2272 

3859 

5445 
7030 



0843 

243 J 

4017 

5603 
7189 



5085 
6693 
8300 



3"7 
4722 

6326 

7930 

9533 



1135 

2737 
4338 

5938 
7538 



9*38 

0737 
2335 
3932 

5529 



7126 
8722 
0317 
1912 
3506 



6692 
8284 

9876 
1467 



3058 
4648 
6237 
7826 

9415 



H^ 



)V 



6373 

79541 
9535 

2695 



8615 
0199 

1783 
3366 

4949 



8773 
10358 

1941 

3525 
5107 



6531 
8112 
9693 
1273 
2853 



6689 
8270 
9851 
1431 
3011 



ttm' 



iJA 



I002 
2589 
4176 
5762 

73^1 



893a 
OJ16 
2100 
3683 
5265 



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160 



159 



6847 
8428 
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3169 

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160 

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275014393327 

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S3 
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57 
58 
59 



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3643 



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4906 5q64]5 222 

64846^426800 



9^399797 



4401216 

2792 
4368 

5943 
75^7 



8220 



12760 
61 
6z 

63 
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66 

67 
68 

69 



4410664 
2237 

38.09 



2770 

7» 
72 

73 



76 

7^ 
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1374 
2950 

4525 
6100 



8378 
9955 



1531 

3107 
4683 

6258 



3801 

5379 
6958 

8535 

0M2 



3959 

5537 

7"5 
8693 

0270 



5 I 6 I 7 I 8 1.9 



N. 27500 L, 439 



41 1 6142744432 

5^95*585360,11 

7273I743 "17589 
88511900919166 

04280585.0743 



45904748 



D J^ro 



6169 

7747 
9324 



6326 

7904 
9482 



7674I7832I7989 



1689' 1847 
326513422 

4840I4998 
64156572 



92489406 
0821 0979 



23942551 
39664123 



9563 
1136 

2708 



8147 



20041216212319 
3580*37383895 
5«55;53«3 547o 



6730,6887 
8304:8461 



9720 
1293 
2866 



53805538 



69517108 
8522 
4420092 
1 66 1 
3230 



5^9: 



42804438 
58,526009 



7265 
8679I8836 
0249 

181811975 

33863543 



7423 
8993 



0405 0562. 
2132 

3700 



4798,^ 
636d6 



7932 
9499 



4954 
522 

8089 



74 4451065 



2780 

83! 



a7$q 

9» 
92 

93 

J!4 

9 

9 

92 
98 




2630 

4^95 

5759 
7322 

8885 



4440448 0604 



2786 

435" 

59 "5j 

7479 
9042 



cm 

6679 
8246 



9655^981.2 

1221JI378 



■5 268 
6835 
8402 
9969 

"5341 



2010 

357" 
5»32 
9692 



^S 8252 
8^ 9811 

^4451370 
ZS 292S 

89 4485 



6042 



3727I 
5288 

6^48 



2943 

4i07 
6072 

7635 
9198 



779" 
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2166 2322 



07600917 



3883 

5444 

7004 



?4o8 856^ 



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1526 

3083 
4641 



6198 



75987754 
9"S4t93"o 

4460799 0865 

22642419 



3818 

5372 
6925 

8477 
991 4470029 



PJ23 
1681 



3099 
4664 

6*228 



7580 
9150 

0719 
2288 

585 7 



5425 

6992 

8559 
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1 69 1 



2478 
4040 

5600 

7160 



8720 
Q279 

"837 
323913395 



4797 



4952 



3256 
4820 
6384 
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9878J0035 
i450|i6o8 
302313180 

4752 
6323 



4595 
6166 



1073 

2635 
4196 

5756 
7316 



8876 

0435 
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355" 

5108 



2445 
4014 



7045 
861.9 



0192 
1765 

3337 

49091 
6480 



7894 



7737 

9307 
08761033 



94649621 



5582 

7 "49 
8716 

0282 

1847 



34" 2 

4977 
6541 

8104 

9667 



1229 
2791 



2602 
4" 7" 



8051 



1190 

12759 
4327 



5738 
7306 
8872 
0438 
2004 



3569 

5"33 
6697 



9823 



6353 6>o9j6^5 
7910 8065.8221 
94659621977^ 
1020 1176 1331 



5912 

7472 



1385 
2947 



5895 
7462 
9029 

0595 
2160 



0901 1058 



2477 2635 

40534210J 

56285785 

72027360 

87768933 



0349 
1922 

3494 
5066 

6637 



82088365 

9778I9935 
1504 

3073 



"347 
2916 

44844641 



3725 
5290 

6853 



82608417 



9979 



6052 
7619 
9185 
0751 

2317 



3.882 
5446 
7010 

8573 
0136 



43524508 



9032 



6068 
7628 



9188 



0590I0746 
21492305 
37063862 
52645419 



9932 
"487 



"54" 
3" 03 
4664 

0224 



77847940I8096 



68206976 
8376 



9343 
0902 

2460 

4018 

5575 



7"32 
853218687 
0087J0243 



1698 

3259 
4820 



0507 
2080 

3652 

5223 
6794 



6209 

9342 
0908 

2473 



4038 
5602 

7166 
8729 
0292 



I 



3, 

3 

4 

S] 
6 

7 
8 



"57 



I854J 

34" 

4976 



638o'6536 



9499 
1058 

2616 



S73i 



9655 
"2 1 41 

2772 

4" 74 43 29 



5886 



7287 
8843 

0398 



1642 17981 1953 2109 



7443 
8999 

0554 



2575i273o' 28g6, 304" | 3"97 ' 3352!3507i 366j 

3974J43f9J4284j4440 4595l4750,49o6l5o6iJ52i6 



5527j5682j5838,;993 
708072351739017546 
863 2'8788,S943 9098 
o 1 84 0339^049410650 



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6i4B|6304!6459i66i46769 
770i;7856j8oi I J8167 8322 
925319408^9563^7199874 
0805 09601 111511270 1425 

~s~i~6~i 7. hn"^ 



156 



158 

Id 

3» 

47 

63 
9$ 

XII 

126 

14a 



"57 

3« 

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156 

16 

31 

47 

6% 

7» 

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lop 

l»5 

X40 



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N* 28000 L. 447 OF Numbers, 



N 



2800 
01 
02 

04 



4471580 
3131 
4681 
6231 

7780 



^5 

06 

07 
08 
09 



2810 
II 
li 

21 

16 

18 
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2820 
21 

22i 

2 



I 



26 

27 

i2 



9329 
4480877 

2424 
3971 

55«7 



7063 
8608 

4490153 
1697 

3 Hi 



4784 

0327 

7868 

9410 
4500951 



2491 

4031 
SSJo 
7109 

8647 



4510185 
1722 
3358 

4794i 
6329 



2830 

31 
3i 
33 
34 



36 
J7 
38 
39 



2840 

4* 
4i 
43 

44i 



46 

48 
49 



7864I 

9399 
4530932 

2466 
3998 



5S3I 
7062 

8593 
45 36 1 24 

' 1654 



3183 
4712 

6241 

7769 

9296 



N 



4540823 0975 
2349 2502 

3875 4027 
54005552 

6924 7077 



y\ ^ 



«735 
3286 

4836 

6386 

7.9JI 
9483 

1031 

2579 
4126 

5672 



7218 
8763 
0308 
1852 

3395 



4938 
6481 
8023 
9564 
1105 



2645 
4185 

5724 
7263 

t88oi 



0338 
1875 

3412 

4948 
6483 



8oig 

9552 
1086 
2619 
4152 



5684 
7215 
8746 
0277 
1807 



3336 
4865 

6394 

7921 

9449 



1 891 



344» 3596 



4991 



6541 6^96 



8090 



9638 
1186 

2734 
4280 

5827 



7372 
8917 

0462 

2006 

3550 



5093 
6635 

8177 



97189872 



1259 



2799 

4339 
5878 

7416 

8954 



0492 

2029 

35<55 
5101 

6636 



5837 

73^9 
8900 

0430 

i960 



3489 
5018 



8074 
9601 



1128 
2654 
4180 

5705 
7229 

2 



2046 



5H6 



8245 



2201 

3751 
5301 

6851 

8400 



9793 
1341 

2888 

4435 
5981 



9948 
1496 

3043 
4590 

6136 



75277681 
9072 9226 
06160771 
21602315 



3704 



5247 



8331 



1413 



2953 
4493 



3858 



5401 



67896943 



8485 
0026 
1567 



3107 
4647 



60326186 

757077.24 
9108^9262 

0646 0799 
2183 

37>9 

5255 
6790 



2336 

3873 
5408 

6943 



8171 8325 847S 
97059859'ooii 

'2391393*54^ 
2772 29263079 

430544584611 



5990 
7522 

9053 
0583 

2113 



3642 
5 17 1 



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4850 

5767 



1905 



2820 

3735 
4650 



1996 



8425 

9341 
0257 

1172 

2088 



291 1 
3826 

4741 



8516 
9432 

0348 
1264 

2179 



55645656 



3003 
3918 
4833 
5747 



6479 657016662 
1 



4025 
4942 

5859 



66846775 
7600 7692 



8608 

9524 
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1355 
2271 



92 



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2 

3 

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83 



92 



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30943186 



4009 



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4 
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6 

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9 
18 

17 
36 
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55 

64 
73 
81 



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4924 5016 
58395930 



6845 



Id 



Pts 



y i ■■ ■■!■ ■ 

K. 490 L. 690 



OF Numbers. 



N 



I4910 
II 
12 

16 

18 
19 



49001690 1 96 1 12049 

2847 293^ 

373338" 
46194708 

5505 5593 



01 
02 
03 
04 

:i 

07 
08 
09 



4920 9651 



21 
22 

23 

il 

:^ 

27 
28 
29 



4930 

3J' 
32 
33 

36 

37 
38 

39 



14940 
41 
42 
43 



6390 

7275 
8161 

9046 



99300019 



6910815 
1699 

2584 
3468 
4352 



5*35 
6119 

7002 
8768 



69205340622 



1416 
2298 
2»8o 



4062 

4944 
5826 

6707 

7588 



, 9350 
6930231 

nil 

1991 

'2872 

375* 
4631 
5511 



6479 6567 



7364 
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0903 
1788 
2672 

3556 
4440 



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6207 

7090 

7974 
8857 



9739 



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2387 
3269 



4151 
5032 

59»4 
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y6y6 



84698557 



9438 
0319 

1199 

2079 



2960 

3839 

4719 

^ 5599 
6390 6478 



7269 
8149 
9027 
9906 



446940785 

45 ^^3 

46 2541 

47 3419 

48 4297 

49 5*75 



7357 
8236 

9115 

9994 
0872 



1751 
2629 

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4385 
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2138 
3024 
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8338 
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1876 
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6295 

7^79 
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9828 
0710 

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2475 
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5120 
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1287 
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3927 
4807 

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7445 

8324 
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0082 

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1839 
2717 

3595 
4472 

5350 



2227 
3113 



39994087 



5770 



6656 

7541 
8426 

9311 

0196 



1080 
1965 
2849 

3733 
4617 



5500 

6384 
7267 

8150 
9033 



9916 



1681 
2563 



2315 
3201 



4973 
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6744 
7630 
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9399 

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1 169 

2053 
2937 

3821 

4705 



2404 2493 
32903379 
41764265 
50625150 
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77187807 
8603 8692 



9488 
0373 



12571346 
2141 2230 



3026 



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5589 
6472 

7355 
8238 

9121 



0004 



0798 0887 



1769 
2651 



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43274415 
5209 5297 

60906178 
697 1 7059 



7853 



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952696149702 



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1375 
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8822 



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2344 



3224 

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65666654.6742 



7533 
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9291 

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1048 



1926 
2805 
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5438 



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8500 

9379 
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2014 
2892 

3770 
4648 

5526 



1I2I 3 t 4 



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0975 1063 

1857 1945 
2739 2828 
3621 3710 



2581 2670 
34673556 

43534442 

52395327 
6124 6213 



95769665 
0461 0550 



2758 
3644 

4530 
5416 

6302 

70107098 7187 
7895 7984 8072 
8780 8869 ' 



8957 
97539842 
0638, 0726 



31143202 



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391039984086 
4793 48824970 

5677576558545942:6030 
656066496737 6825*6914 
744475327^^2077097797 
832784158503 
92109298 



45034591 

5385 5473 



4680 
5561 



6266163546443 



7148,7236 
8029*8117 



8910 



979019878 



0671 

1551 
2432 



8998 



0759 
1639 
2520 



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419114279 

507 U 159 
5951:6039 

6830I6918 



770917797 
8588 8676 



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0345 



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2318I2407 



3291 
4175 



1611 

2495 

3379 
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8592 8680 



9386 947419563 



026903570445 
1151 12401328 

20342122j22IO 
2916 30043092 
3798 3886.3974 



8205 



4768J4856 

5649'5737 
653116619 



7324I741 27500 



8293:8381 



9086I91749262 
9967100550143 
084709351023 



1727 
2608 



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4367 

6126 
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4455 
5335 



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04330521 



1224^13121399 



2102 
2980 

3858 



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2366 

3244 
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4999 
5877 



2190*2278 

3068J3156 

3946:4034 

47364824'49ii 



570115789 



1815.1903 

2696,2784 



3576I3664 



4543 
5423 



62146302 

7094J7182 



7973,8061 

8764I8852 8940 

97309818 

06090697 



1487 



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1575 



2453 

3331 
4209 

5087 

5964 



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88 



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9 

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70 
79 



87 



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4 
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53 



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6929 



8683 



7017 



78067894798 



8771 



9£6o-9648 



973519823 



556950437I0524 

I3i3:>40i 

2i89'2277 

306513153 
3941 [ 4029 

4817I4904 



06 1 a 0700 



56925780 



6568166556743 
7618 

8493 



7443I7530 
8318:8405 



91939280I9367 
696oo67'oi55 
0942 1029 
1816I1903 
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3564.3651 

4438:4^25 

531 115599 
6185J6272 

7058, 7145 



793»;8o!8 
8So4;889i 
9676'9764|985 



78169705490636 
791^1421! 1 508 



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81 
82 

84I 



86 

87 
88 



6652,6739 
7523J7610 
839418481 
926419352 



89,698013510222 




1005J1092 
1876; 1963 
27462833 



3616 
4485 



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8831 



3703 
4572 



5355 5442 
62246311 

7180 

8049 

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7105 
2 
8859 



1488 
2364 

3240 
4116 



4992 
5867 



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1116 

1991 

2865 



6315 
7192 
8069 
8946 



6403 
7280 
8157 

90341 
9911 



1576 
2452 
3328 
4204 



5079 
5955 



0787 
1663 
2540 

34«6 
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5167 
6042 



68306918 

7705 7793 
8580 1 8668 

945519542 
033010417 
120411291 
2078 2166 
295213040 



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4700I4787 

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6359.644716534 
7232 732017407 
8193)8280 



8105 
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08 II 10898 



1596:1683:1770 



2293J238JI2468J2555I2642 

3165.3253 

4037.4124 
49094996 
57805867 



334013427 
4212I4299 
5083I5170 



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3514 
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7697I7784 
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1180 
2050 
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7267 
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1267 
2137 
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3877 



5616 
6485 



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7000 
7871 

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0483 



1354 
2224 

3094 
3964 



5703 
6572 



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8310 
9179 



6491 
7368 
8245 
9122 
9998 



6578 6666 6754 6842 



0875 

1751 

2627 

3503 
4379 



5255 
6130 

7005 

7880 



0962 

1839 
2715 
3591 
4467 



7456754376317719 
8333 8420 8508 8596 

9209929793859472 

0086 0174 0261 034a 



5342 
6217 

7093 
7968 



8755)8843 



9630:97 1 7 9805 9892 9980 

050410592 0679 0767 0854 

146615541641 

2340I24282515 



1379 
2253 

3127 



32143302 



4001 4088 
4874J4962 

574815835 
6621 

7494 



8367 
9240 
0113 
0985 
1857 



2729 
3601 

4475 
5345 



7582 



8455 
9327 

0200 
1072 

1945 



7 



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105011138 1225 
19262014)2102 

2802 289012978 

3678 3766J3854 
455446421 4729 



54305517 
6305 6393 

71807268 

80558143 

89309018 



5605 
6480 

7355 
8230 

9105 



3389 



41764263 

5137 
6010 



5049 
5923 



6709 6796 6883:6970 



76697756 



8542 

9415 
0287 

1160 

2 032 



1728 
2603 

3477 



4350 
5224 

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8629 
9502 



7844 



8716 
9589 



03740462 



1247 
2119 



1334 
2206 



6216^303 



28172904I2991 
3689 3776J3863 
456046474735 

54325519 



6390 



5606 5693 
6477 6565 



7087 717417261 7349J7436 
79581804581.3282208307 
8829 89io|90o3^909o;9i77 
9700 9787 J9874 996 1 J0048 
0570{0657lo744}o83 1 091 8 



3078 

3950 
4822 



1441 
2311 



4051 
4920 



152811615 



2398 



318113268 



4138 
' 5007 



57905877 



6659 6746 6853 6926 7007 



75287615 



8397 



9266 2253 
6 



84848571 



2485 
3355 






I 7 



170211789 
2572I2659 

3442*3529 



42244311 
50945181 



59646050 



7702 7789 



8658 



943995269613 



8 



4398 
5268 



6137 



7876 
8744 



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D 



88 
9 

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S3 
6x 

70 

9\ 79 



87 



9 

17 
16 

S5 



70 

78 



Pts 



' m V. 



N. 510 L. 7oy 



OF Numbers. 



(89) 

D Pro 



"N^ 



5100 
01 

02 
03 

04 



7075702 

6S53 

7405 
8256 

9107 



5787 
6638 

7490 

8341 
9192 



2 I 3 



07 
08 
09 



5110 
II 
12 

13 

HI 



•A 

«7 

18 

«9 



5120 
21 
22 
23 

\l 

27 
28 

29 



995710043 

7080808,0893 

1659 1744 

2509 2594 

3359 3444 



4209 



5872*59576042 



6724 

7575 
8426 

9277 



0128 
0978 
1829 
2679 

3529 



4379 



4294 

5059514415229 

5908:59936078 



6758:6843 
760717692 



8456 
9305 



70901540239 



1003 
1851 



2700 

3548 

4396 

5244 
6091 



5130 

31 
32 
33 
34 



36 

37 
38 
39 



5140 

41 
4* 
43 

il 

45 
46 

47 
48 
49 



6939 
7786 

8633 
9480 

7^00327 



8541 
9390 



lobS 
1936 



2784 

3633 
4481 

5328 

6176 



6928 
7777 



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2020 

2866 

J 



7023 
7871 
8718 

9565 
0412 



8626 

9475 
0324 

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2021 

2869 

37«7 

4565 

54»3 
6261 



1258 
2105 
2951 



37«3i3797 
4559^4643 



5404I5489 
62506335 
7096 



7108 

7955 
8803 

9650 

0496 



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2189 

3056 
3882 

4728 



6809 6894 



7660 
8511 
9362 



0213 
1063 
1914 

2764 
3614 



4464 

53H 
6163 

70»3 
7862 



8711 



95609645 



0409 
1257 

2I06 



2954 

3802 
4650 
5498 

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7193 

8040 

8887 

9734 
0581 



794* 
8-^86 



963' 

7110476 

1321 
2165 

3010 



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3854 
4698 

5542 
6385 
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7180 
8026 



1428 
2274 
3120 
3966 
4812 



557456585743 
64191650416588 

7434 



7265J7349 
8110:8195 



8871J8955 I 90409124 



9716980019885 



0561 
1405 



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1659 
2503 

3347 



14901574 



2250.2334.2419 
309413178.3 



263 



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4782;4867;495i 
5626157 I0J5795 
647065546638 



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1 148 
1999 
2849 
3699 



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5399 
6248 

7098 
7947 



8796 



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1342 
2191 



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5583 
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7278 
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3205 

4051 

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5035 

5879 

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7566 



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69797064 



7830 
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19532 



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8766 

9617 



0383 

1233 
2084 

2934 
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7H97234 
80008085 

8851 8936 

9702*9787 



0468 

1318 
2169 

3019 
3869 



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5484I5569 

6333.6418 
7183I7268 
8o32i8ii7 



8881 
9730 



0579 
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2275 



3972 
4820 
5667 



8966 

0663 
1512 



7 I 8 



0553(0638 
1403 1488 
22542339 
3IO43189 

3954i4039 



6468 

7319 
8171 

9022 

9872 



0723 

1574 
2424 

3274 
4124 



480448894974 
56545739I5823 
6503^658816673 

7352743717522 
8202 8287'837i 



9051 91 3619220 
9900 998410069 
0748!o833!o9i8 

1597] 1682! 1766 



236012445125302615 



31243209 



4057 
4904 

5752 



651516600 

7362J74477532 
8210I82948379 



9057 
9904 

0750 



9141 
9988 
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2443 
3290 

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0073 
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1682 
2528 

3374 
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1766 
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3459 
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58275912 

6673,6757 
75187603 

83648448 

9209,9293 



0054 
0898 

1743 

2587 

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0138 
0983 
1827 
2672 
35*6 



4445 



427614360 
51205204I5289 
59646048 ' 
6807 6892 

t^ 7ns 



TT 



329313378 



4141 (4226 

4989I5Q74. _ 
583715922:6006 

66846769i6854 



3463 
4311 

5159 



7617 
8464 

9311 

0158 



7701 
8548 

9395 
0242 



1004 1089 



5996 
6842 
7687 

8533 



1851 

2697 

3543 
4389 
5235 



1936 
2782 
3628 

4474 
5320 



6081 

6927 

7772 
8617 



93789462 



0223 
1067 
1912 
2756 
3601 



0307 
1152 
1996 
2841 

3685 



6166 
7011 
7856 
8702 

9547 



0392 
1236 
2081 
2925 

3769 



4529 

^ , 5373 
61326217 

6976I7060 
78197904 

7 8 



4613 

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(90) 



Lo G ARIT H MS 



7118072 
8915 

9759 
7120601 

1444 



2287 



8157:8241 
90009084 
9843I9927 



0686 
1528 



2371 



3129:3213 

39714056 
48 1 3 4898 

5655,5739 
64976581 

73397423 
81808264 

902x9105 

9862*9940 

r 



0770 
1613 



3 I 4 



832518410 

91689253 
001 X 0096 
08540939 
1697 1781 



2455 
3298 

4140 

4982 
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7 1 3070310787 
15441628 
2385J2469 

3225I3309 
40654149 

4989 



2539 
3382 
4224. 
5066 
5908 



6665 

7507 
8348 

9189 

0031. 



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1712 

2553 
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6750 

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5829 



5073 
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5745 
6585 

7425.. .,.,,- 
826483488432 



66696753 
75097593 



0956 
1796 
2637 

3477 
43'7 



5^57 

5997 
6837 

8516 



91049187192719355 
9943|oo27'oi 100194 
714078208600949 1033 
1 620' 1 704! 1788 1872 
2459.2543I26272711 



6834 
7675 

8517 

9358 
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1040 
1880 
2721 

3561 

4401 



5241 
6081 
6921 
7761 
8600 



3298:3381134653549 
41 36 422o;4304 4387 

4974:5058,5 H2I5226 

5812 589659806063 



6650 



7488 



6734.6817 



757117655 



83251840918493 

9162I92469330 
7 15000010083.0 1 b'] 

083710920 1004 



6901 



9439 
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1117 

1956 

2795 



3633 

447 « 

5309 
6147 

6985 



77397823 
85768660 



^674117571^841 
2510:25942678 

3347I3430.3514 
4183I42674350 

5019:51035187 



N 



5856 
6691 

7527 
8363 



59396023 
6775.6859 



94 H 9497 
025 1 '0335 

10881171 



1925:2008 
27612845 
3598:3681 
4434,45 » 8 
52705354 



9337 
0180 

1023 

1865 



2708 

3550 
4392 

5234 
6076 



84948578 



6918 

7759 
8601 

9442 
0283 



1124 

1964 
2805 

3645 
4485 



5325 
6165 

7005 
7845 



9523 
0362 

1201 



2878 



4555 

5393 
6231 

7069 



7906 
8744 
9581 
0418 
1255 



61066190 
6942!7026j 
7611 {7694 777817^1 

8446853018613:8697 



919892829365 



1"rf 



9449<9^3 



z 



2092 
2929 

3765 
4661 

5438 



6273 
7109 

7945 
8780 

9616 



N. 515 L 



9421 



8663 
9506 



0264 03^ 



1107 
1950 



2792 
3634 

4477 

53«9 
6160 



7002 

7843 
8685 



95269610 
03670451 



1208 
2*048 
2889 

3729 



5409 
6249 
7089 
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8684 8768 



9607 
0446 
1285 



20402124 



2962 



3801 

4639 

5477 
6315 

7*53 



8 



1191 
2034 



8747 
9590 

0433 
1276 

2118 



2876 

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5403 
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7928 
8769 



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2132 
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18 



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56545728 

(53896463 
7124 7197 
78587932 
8593 S666 



0061 

0795 
1529 

2263 



0135 
0869 

1603 



2924)2997 



3731 
4464 

5197 
5931 



659066646737 



7397 
8129 

8862 
9595 



02540327 



1060 
1792 

2524 
3256 



6768 6841 
74997572 



46464719 

5378545' 
61096183 



8303 

9034 
9765 

0496 

1226 



6914 
7645 



8376 
9107 

9838 



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2687 

3418 

4148 

4878. 



2030 
2760 

349» 
4221 

495 » 



7 



9 I 



9183 
9918 
0654 
1390 
2125 



2861 
3596 

433J 
5066 

5801 



6536 

7271 

8005 
8740 
94019474 



0208 
0942 
1676 



23372410 
30703144 



3804I38 

4537 

5271 53441 
60046077 



3877 
4011 



6810 



74707543 
820318276 

893519009 

9668.9741 

0400,0474 



1x33 1206 
1 8651938 



2597 
3329 



39884061 



4793 



6987 
7718 



2670 
3402 

4>34 



4866 



55245597 
62566329 



84498522 
91809253 



9911 



05690642 
1299 



1372 



2103 
2833 



4294 
5024 



8 



7060 

779* 



9984 
0715 

1446 



2176 
2906 



35643637 



4367 
5097 



73 



5 



5 



73 



iaip 



DjPts 



('06) 



Log A/kiT h 



N 



5950 
5» 

S3 
54 



11 

57 
58 
59 



59i6o| 
01 

64 



6S 
66 

67 
68^ 

69 



S970 

7» 

7» 

73 

24 

7<5 
77 
; 78f 
, 79 



5980 
81 



I 

1 



82 

86 
87 

8a 
89 



7745170 

6629 

7359 
8088 



8818 

9S47 
7750276 

1005 

»734 



2463 
3191 
3920 
4648 

5376 



01Q4 
6832 



75607633 



8288 
9016 



9743 
776047 

^ 1198 

1925 

2652 



3379 
4106 

4«33 

|S59 
6286 



7012 

8464 
9190 
9916 



2093 
2818 

3543 



I 



|599<^ 

9^ 
92 

94l 



96 

97 
98 







8891 
9620 

0349 

1078 

1807 



89649036 



9^93 
0422 

1151 
1880 



2535 
3264 

3993 
4721 

5449 



6177 
6905 



2608 

3337 
4065 

4794 
5522 



M/l 



N.S95L.77+ 



4 



9766 
0495 
1224 
1952 



2681 
3410 

4^38 
4867 

5595 



8^61 



6250 
6978 
7706 



90S99161 



98x6 



^323 
7051 

7779 



9109 
9839 
0568 
12971 

2025' 



2754 

3483 
4211 

4939 
5668J 



5 5^8 
2646337 

6994I7067 

7724J7797 
8453I8526 



9182 
9911 
0641 
1369 
2098 



2827 
3555 



9255 
9984 

0713 

1442 

2171 



932894019474 
005701300203 
0786 085910932 
1515 1588 1061 

r2244i21I7 239Q 



29002973 



3628 



428414357 
50x215085 

5740^581 3|5886|5959 



6396' 
7124I 

7851! 



843485068579 



9234 



0543 06x6106890762 



1 27 1 
1998 
2725 



9889 9962 



3452 

4179 
4905 



U43 
2071 



1416 
2x43 



2798 2870 



3524 



3597 



5^32 5704 
63586431 



425 '14324 
49785051 



6503 



7084I7157 
78117883 . 
853786098682 

9263:93359408 
9988'oo6i!oi34 



9307: 



6469J6541 
7196J7269 

7924I7997 
865218725 

9380I9452 



0034, 



1489; 
22x6 
2943 



Ioio7jOi8o 
'o834'o907 



3670^ 

4397J 
5123; 



1562 
2289 
30x6 



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4469 



1634 
2361 

3088 



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4542 



5777 5850 



6576!!! 



5x9615269 

5922I5995 
664916721 



723017302; 
7956 80281 



8754!'8827l890o 

948o!;9>53 
oao6,:0279i035i 



777064207140787 



2165 
2890 



X5X2 
2238 
2963 



36x6.3688 



42684341:4413.4486 
499350665x38)52x1 
5718579x586315935 
6443 65x5165886660 
7x677240,73x217385 



0859I 
1585 
23x0 
3035 

3761 



73757448 

I8101 8F74I8246I83 



c6Si 
6410 
7x40 

7869^94 



8599 



370X 

4430 
5x58 



6614 

7342 
8070 



8798 8870 8943 



9525 



0253 
0980 
X7a7 

2434 
3161 



3888 
4615 

534* 
6068 



7520 



8972 

9626 9698 

0424 



0932 
1657 
2383 
3108 

3833; 



m 



7892 
8616 

9340 
7780065 

0789 



7964 
8689 

94«3 

0137 
086 X 



803718x098x821 

876188348906 

9485 

0209 

0933 



0282 
X006 



4558! 
5283 
6oq8 

6733 
74S7 



1004H077 X149 

fc730ii8o2 1875 

2455U5282600 

i3>8xi3253!33.26 

39o6U978 !405 



535<^ 



95589639 

0354 
X078 



Ll± 



75:30 



5428I 



463i!4703!4776 
5501 

6225 

695.0 

I7675 



6o8o{6x53 
6805,6878 



8 



72x3 
2 



57545*27 
64836556 

7286 

80x5 



8672I8745 



4502 
523X 



30463x18 

37743847 



4575 
5304. 

6032 



66876760 
7488 



7415 
8x43 



8215 



9598 9671 



0325 

1053 

1780 
2507 



323433 



3961 
4687 

541^ 



61406 



6794 6867 6939 



7593 

»9 
9045 

977^ 
0496 



X222 



2673 

3398 
4x23 



4848 

5573 



7602 



82548327 
89789051 
9703 9775 
0427I0499 



115X 



1223 



8399 
9123 

9847 

0571 

1295 



8471 
9x96 



0398 
1x25: 

1852 



4033J 
4760 

5486, 
JK13 



7665 
8391 
9117 

9843 
0569 



X295 

19471202a 



2745 

3471 
4x96 



49XX 
5646 



6298 63.70 
702.2 7095, 

7747 78i9 



1368 



8 



8544 
9268 



99^0 9^99^ 
06440716 



1440 



I* 



73 



71 

XX. 

37 



66 



72 



7 
141 

291 

43 

50 



15 



S 



N. 600 L. 778 



OF, Numbers. 



TT 



6000I7781513 
2256 
2960 

3683 
4407 



09 



19 



29 



TT 



5130 

5^53 



657666496721 



7299 
8022 



8745 
9467 

7790190 

0912 

1634 



7810369 
1088 
1807 
2526 

3245 



3963 
4681 



'5851 
2309 

303* 
3756 

4479 



5202 
5926 



737* 
8094 



8817 

9540 
0262 

0984. 

170^ 



2429 
3>50 
3872 

4594 
5316 



3H5 

39^5 
4685 

5405 
6125 



6845 



8284 
9003 
9722 



0441 
1160 

1879 
2598 
3316 



4035 
4753 



540015472 
6118 



6190I6261 

683616908,6979 



1657 
2381 
3105 
3828 

4552 



5275 
5998 



7444 
8167 



8889 
9612 

0334 
1056 

^779 



2501 
3"3 

3944 
4666 

5388 



97«5 



1156 
1877 

«597 



[8356 
9075 

9794 



P513 
1232 

1951 

2670 

3388 



4107 
4825 

5543 




5347 
6070 

6793 

7516 

8239 



5419 

6143 
6866 

7588 
83«» 



8962 



9034 



96849756 
0479 
1201 

»923 



»573 

3295 
4017 

4738 
5460 



6181 
6831(6903 

75527624 

8*73 8345 
8994 9066 



9787 



04350507 



1228 

»949 
2669 



69176989 



7564763677087780 



8428 

947 
9866 



0585 

13044 

2023 

2742 

3460 



4179 
4897 



6253 

7696 
84x7 

9138 



3389 
4109 

4829 
oz^9 Q34' 



7061 



8500 
9219 

9938 



4250 
4969 



5615I5687 
6333 6405 



7051 



7J23 



7047 
7768 

8489 

9210 



»947 
2670 

3394 
4117 



4841I4913 



7 



2019 
2743 



8 



2092 
2815 



34663539I3611 
4190 



54921 
6215 

6938 

7661 

8383 8456 



5636 



63596432 



9106J9178 
9820 990 X 



055 X 

1273 

>995 



925 X 

., 9973 
0623 0695 



2717 

3439 
4161 

4883 



5604 5676 



63266398 



993 « 
0652 

1372 
2093 
28x3 



3533 
4253 
14973 
5693 



7133 
[7852 
857X 
9291 



0729 

1448 
2x67 

2885 



»345 
2067 



2790 

4233 
4955 



71x9 

7840 
856X 

9282 



0003 
0724 

>444 
2x65 

2885 



7204 

7924 
8643 

9363 



0010,0082 



080X 
X520 
2238 
2957 



360413676 



4322 
504X 

US759 



4394 



7082 
7805 
8528 



5709 



Fro 



2164 
2888 



7i55 

7877 
8600 



2862 



35843656 



4305 
5027 

5748 



6470 
719X 
7912 

8633 



935494269498 



0075 
0796 
15x6 
2237 

2957 



3(>77 

4397 
5x17 

5837 



9323 
0045 

0768 

X490 

2212 



2934 



4377 
5099 

5821 



6542 
7263 



7984 8057 



8705 



047 
0868 

15S8 

2309 

3029 



3749 
4469 



5909 



655766296701 



7348 
8068 

8787 



0x5410226 



087310945 
X592 1663 
23x02382 
3029 3 xox 

374838x9 



44664538 



9395 
01x7 

0840 

1562 

2284 



3006 
3728 

4450 
5x71 

5893 



6614 
7335 



8778 



02x9 
0940 
x66o 

2381 
310X 



3821 
454« 



5x895261 



598 X 



7420 
8140 
8859 



95069578 



0297 



5XX2J5X84I5256 



583x!5902'59746o46 



6477.6549-66206692 



7x9517267 



7338J74J 

7 



4610 
5328 



6764 
7482 



7* 



(io8) 



Loo AS.IT H M S 






050 

53 

54 



7817SS4 
8272 

8989 

9707 



56 
57 
58 
59 



7626 

8343 
9061 

9778 



7697 
8415 

9133 
9850 



7820424 0496 

Ii4i|i2i3|i2d5 
i359!i930j2002 



7769 
8487 
9204 
9922 



0568 0639 



2576 

3293 
4010 



6060 
61 
62 

h 
65 

66 

67 
68 

69 



J357 
2074 

..... 2791 
3364I34363508 

408 1 Ui 53 422s 



2647:2719 



4941 
5658 



4726479814870 
5443 55i4;|586 ^ 
6i59;623ij6303'63746446 
6876:6947 j70i 9 
759276647735 




7091 7162 
7807 7878 



8308183801845 118523:8594 
90249096:91671923919310 
9740I98 1 219883 995 5 10026 



7830456 
1171 



6070 

71 
7« 
73 
74 



76 

77 
78 
79 



6080 
81 
82 

83 
84 



o527;o599|o670 
124313141386 



1887 195812030 

2602 

33i8 3389;346i 

4033 
4748 



5463 
6178 



7607 
8321 



2102 
2817 

3532 



7984I8056 
8702J8774 
9420I9491 
0137(0209 



0855 



1572 



22892361 



3006 
37*3 



4440451 X 



1644 



3078 



N. 605 L. fSi 



8 



8128 
8846 

9563 
0281 



9 IDlPro 



0926 0998 



8200 
8917 

9635 
0352 

1070 



1715 
2432 

3H9 



3794 3866 



4583 



5156 

58o'!S873.,.. 
6518 65896661 6732 



7234173057377 
7950;8o22!8o93 



522II5300I5371 

594460166088 

6804 

7449 752^ 
81658236 



0742 
1458 



41041417614217 
4819I4891I4962 



2173 
2888 
3604 

43«9 
5034 



5534|56o6!5677i5749 
6249 6321 6392 6464 



689269641703517107 7178 



8666|8738 8809 
93829454.9525 
00980169,0241 



0814 
1529 



0885I0957 
1601J1672 



2245 
2960 

3675 
4390 



2316I2388 
3032]3«03 

374713818 
4462I4533 



1787 
2504 
3221 
3938 
4655 



o3"3 
1028 

1744 



8881 8952 

9597 9^8 
0384 
lioo 
1815 



i5«o5:5>77 



7878177501782117893 
839358464185368607 



9036J9io7J9i79,9250 



85 
86 

87 
88 

89 



6090 

9« 
92 

93 
94 



9750 
7840464 
1178 
1892 



9821 9893 
0536:0607 
1250*1321 
1963'2035 



2606 



. , 9322 
9964:0036 

067810750 

13921464 



267712749 



•0107 
0821 

2I06'2I78|!2249 



58205892 
65356606 
72507321 
7964 8036 

186798750 



5248 



2459 

3175 

3890J 
4605 

5320 



5963 
6678 

7393 
8107 

8821 



2531 

3246 

39^ « 

4676 

539» 



6o35j6ip6 
6749.6821 



7464 



8179^8250 



8893 



9393i9464'9536'96o7 



7536 



8964 



2820 



6173 
6886 

7599 

8312 

9024 



95 
96 

97 

98 

99 



N 



9737 
785045c 

1162 

1874 

2586 



3319339^1346213534 

4033 
4746 

5460 



4104:4176 
4818:4889 
5531I5602 



4247 



2891 
3605 
4318 



49605032 
567415745 



6244I6316I6387J6458 
6957j7029;7 1007171 
7670i7742j78i3'7884 
8383;8454l8526|8597 
9096'9i67'9238;93io 



0179 
0893 
1607 
2320 



0250 
0964 
1678 

2392 



0321 

»035 

»749 
2463 



2963303413105 

^7637483819 
439044614532 

i5»03'5»74|5246 
58165888,5959 



0521 
1233 



980S 9880995 1062 2 
0592'6663 0735 

i304>376;i447 
194512017:208812159 

2658;2729l28oo;287 1 



I ' 2 I 3 I 4 



165 29:660116672 
I7242I73147385 
7955(80278098 
8668;8739i88ii 



9381I9452 



0093 
.0806 
!i5«8 
12230 

2942 



0165 
0877 



9523 



9679 
0393 
1107 
1821 

2534 



31773248 
38903962 
4604 4675 
53175388 
60306102 



6743681 
7456 75*1 



0236 
0948 



158911661 



2301 



2373 



30143085 



I 



8169 
8882 

9595 

0307 
X019 

1732 

2444 
3^56 

r 



8241 

8953 
9666 



0378 
1091 
1803 
2515 
3227 



JL 



72 



D 



N;6|oL. 785 



OF Numbers. 



"K 



6100 
01 
02 

03 
041 



7853<98 3370 
40104081 

47*^ 4793 
6216 



07 
08 

09 



(Siio 
11 
12 

«3 



7860412 
M23 
1833 

*544 
3^54 



15 
16 

»7 
18 

>9 



27 
28 

29 



6130 

31 

3* 
33 

341 



36 

37 
38 
39 



5434 
6145 



6857 
7568 
8279 
8990 
9701 



6928 6999 



7639 
8350 

9061 

9772 



3965 

4675 

5385 
6095 



0483 
1194 
1905 
2615 

33£S 



40364107 
47464817 

6166:6237 



6805 687616946 



6i2oi 7514 

21 8224 

22 8933 

23 9643 

^4 787035^ 



1061 

1770 

2479 
3188 

3896 



4605 

S3»3 
6021 



8146 
8854 

7880269 
0976 



^140 

41 

42 
43 

44 



4S 

47 
48 
49 
N 



1684 

?39' 

3098 

3805 
4512 



8045 



344« 
4153 



4864 4936 



5576 
6288 



7710 
8421 
9132 

9843 



0554 
1265 

1976 

2686 

3396 



3512 



42244295 



5647 



63596430 



7070 

7781 
8493 
9204 

99«5 



0625 

1336 

20471 

2757 

3467 



4178 
4888 

5598 
6308 

7017 



758576567727 
8295.83668437 
9004 9075 19 146 

97*4978419855 
04231049410565 



1132 
1841 



1203 
1912 



25502621 

3258:3329 
396714038 



4676147464817 



5384I5455 
6092:6163 



67306800I6871 
7438 



7509:7579 



62346305 
69427013 

76507721 



8216:8287 
8924*8995 
9632:9703 
03400410 
I047{i]i8 



175411825 
2462I2532 
3169:3240 

387613947 
458314653 



5219 
5926 
6632 

73397409 



5290:5360 
5996 6067 

6703 6773 



8116 



7480 
8186 



1274 
1983 
2691 
3400 
4109 



5526 



3583 



5007 
5718 



7141 
7852 

8564 

9275 
9986 



0696 
1407 
2118 
2828 

3538 



4249J 

5669 

6379 
7088 



36543726 



4366 

5078 

5789 
6501 



4437 

5H9 
5861 

6572 



7212:7283 
792417995 



8635 
9346 
0057 



8706 

9417 
0128 



0767 0839 

1478 «S49 
218912260 

2899:2970 
3609I3681 



4320J4391 
;503o|5ioi 

J5740j58ii 
164506521 



7798 
8508 
9217 
9926 

0635 



7«59 



7869 



7230 



7940 



«34S 

2053 
2762 

347" 
4180 



857918649 
9288J9359 
99970068 

0706 0777 



14151I486 
2 1 24 2 1 95 
2833.2004 
3542I3613 
42504321 



4888 
5596 



3797 

4509 
5220 

5932 
6643 



3868 
4580 
5291 
6003 
6714 



7355 
8066 

^777 
9488 

0199 



0910 
1620 
2331 

3041 
3752 



4462 

15"72 
5882 
6592 
7301 



8011 
8720 

9430 
0139 

0848 



I 



2266 
2975 



4392 



1 1 



8358 

9066 



97749844 



0481 
1189 



1896 
2603 
3310 
4017 



543 « 

6138 

68446915 
755» 



8429 
9M7 



49595030 
5667I5738 



637664466517 



7084 
7792 



J8500 
19207 



7"55 

7863 



9x01 

5809 



7225 
7933 



0552 
1259 



1967 
2674 

3381 
4088 



472414795 



5502 
6208 



8257 



7621 
8327 



199 1519986 

; 062 3 [0693 

140X 



8570*8641 

9*78.9349 
0057 



7426 
8137 
8848 

9559 

0270 



0981 
1691 
2402 
3112 

3823 



4533 

5243 

5953 
6663 

7372 



8082 
8791 
9501 
0210 



39391 
4651 

5363 



7497 
8208 

8919 

9630 

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50185071 



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21 
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27 
28 

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7133 7i«6J7239b29i|7344 

7661 77147767 7820 7873 

^19^243^29516348 8401 



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8774 8827 

935.6 



8350! 
8880 

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9886 9939 

0415 0468 



P786 3839 0892 0945 3998 
1315 1308 14ZI 1474 1527 
1844 1897 1950 2003 2056 
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3008 3061 3114 



4066411914172 
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9775 ^28fa88o ^33 9986 

^150305 9354M09 ^* 05 H 
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135914121465151715701 
1887 1940 1993 2045 2098 
2415 24682521 2573 2626 
39432996304331013154 
347' 3553 357^6^^682 



4051 



4104 4157 4209 4262 43 15 4368 4420 4473 



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1380 



1907 
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1959 
2486 



19603013 



3539 



2012 



4013 4066 41^18 4171 



88508903 



3643 



6267 S3 20 



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30653118 

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20652117 



2644 
5171 

3697 
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9992 0045 0098 



0521 0574 



8403 

8933 
9462 



2109 
2638 



8456 8509 
89869039 
95159568 



0627 



105011103 1156 1209 
1580 1653 1686 1738 
2162 2215 2268 



3167 3220 



2691 2744 2797 



3»55 



369637493802 
422542784331 
4754480748604912 

5?83lS335 53885441 



5811 586459175 970 6 023 



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6869 6921 6974 7027 7080 

7397 7450 75<>3 7556 7609 
7926 7978 803 1 8084 3 1 37 
8454 8507 8560 8613 8665 



1095 



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4790 -^84314895 4948J5001 

5317537054235476552'' 
5845 5898 59506003 605 

6372 6425 6478 65311658 



6900 



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3273 



1148 



2732 



3207 3260 33 1 213365 
3734 378713*4op893 



6952 
7480 



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0590 0642 0695 
1116 11D9 1222 



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567 d62o 3673 3725 3778 



1 201 



1623 1676117291781 1834 
2151 22042257 23092302 



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2321 
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33263379 



3908 



4384 4437 
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7058 
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8534 8587 



91 14 9167 



2170232312275 
2697 274912802 
3223 3276 
3750 JS02 
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4382 



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4026 

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615862106263 6315 
6683 6735 6788 6840 



7260 



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9884 9936 



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2506 



3030 

3554 
4078 

4602 
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4162 



5213 
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4750 
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29 

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7381 

7907 

8433 

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1010 1062 

1535 1588 

2061 2113 



2639 

3164 
3690 

4215 



4687 4740 



5265 
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7837 7890 
83628415 
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9989 



6645 



7640 7693 



71697221 



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56505702 



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7745 



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9316 



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1562 
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53295382543454875540 
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6382 6434 6487 6539 6592 
6908 6960 7013 706617 II 8 



743474867539 
796080128065 
8486 8538 8591I8644 



90129064 



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0589 0642 0694 0747 



9538 



1640 
2166 



2691 

3217 

3742 



4267 4320 
4793 4845 4898 



53« 

5843 



85370 



7942 
8467 



3187 
3711 

4235 
4759 
5283 



7486(7539 7592 7644 
81188170 



9S90 9^3 



1167 
1693 
2218 



2744 



3269 3322 



3795 



5895 



7995 
8520 



05660618 
10901143 
1614 1667 
21392191 
26632715 



5807 
6331 

6855 

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8426 
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4288 
481 



5860 



7431 



7954 



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3847 
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91699 



9695 



1220 1272 
1745 1798 
22712323 



2796 2849 



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5948 6 



368 6420 6473 6525 6578 
893 6945 6998 7050 7103 

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8152 
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0094I0 1 46|o 1 98 025 1 



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1195 

1719 

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32403292 
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4340 
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53365388 



5912 



690769607 



8478 
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8007 



4425 
4950 



8696 



9748 



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3900 3952 



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4393 
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4477 
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5528 
6053 



1300 
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2820 2873 



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4445 
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5493 



59646017 

6383JS436|6488k54i 

012I7064 

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OF Numbers. 






8300 
01 
02 
03 
041 



9190781 
1304 

1827 



Q5 
06 

07 

08 

09 



8310 
II 
12 

»3 
JM 

16 

»7 
18 

8320 
21 
22 

23 
24J 



25 
26 

27 
28 

29 



23502403 



2873 



39«9 
4442 

4965 

5488 



0833 
1356 



1880 1932 



2926 



33963449 



3972 

4494 
5017 

554^ 



60106062 



6533 

7S7^ 
8100 



8623 

9H5 

9667 



9200189 0241 



0711 



»233 

«7S5 
2277 

279^ 
3321 



3842 
4364 



8330 

31 
32 

33 
341 



35 
36 

37 
38 
39 



9056 

9577 
9210098 



8340 

41 
42 
43 
44 

46 

47 
48 

JH 
N 



[ 



5407 
5929 



6450 
6971 

7493 



8535 



1661 
2181 
2702 
3222 

3743 



6585 
7108 

7630 
8152 



8675 
9«97 



0763 



1285 
1807 
2329 
2851 

3373 



3895 



48864938 



5459 
5981 



08860938 



1409 



H55 
2978 



3501 
4024 

4547 
5069 

5592 



6115 



7160 



8205 



8727 



9719 9771 



0294 



860 
2381 
2903 

3425 



1 461 
1984 
2507 



3030 3083 



3553 
4076 

4599 
5x22 

5644 



6167 



6637 6690 6742 



7212 



76827735 



8257 



8779 



92499301 



0990 
1513 
2037 
2560 



3606 
4128 
4651 

5»74 
5697 



6219 



7264 

77^7 
8309 



9824 
0346 



08160868 



8831 

9354 
9876 

0398 

0920 



1390 
1912 



2434 2486 



3947 



4416 4468 



499<^ 
6033 



6502 
7023 

7545 



655466066659 



80148066 



8587 



9108 



962^9681 



9160 9212 



01 50 0202 0254 0306 



4263 
4784 



6345 



3274 
3795 



4315 
4836 



53045356 
58245876 



6397 



70767128 



7597 



811881708222 



86398691 



0619 0671 0723 0775 
1140 1 192 1244 1290 



17131765 
22332285 
27542806 



33*7 
3847 



2955 

3477 



3999 
4521 

5042 

5564 
6085 



7649 



9733 



1817 

2337 
2858 

3379 
3899 



4367 
4888 
5408 
5928 



4420 



5460 



1442 
1964 



3008 
3529 



4051 

4573 

5094 
5616 

6137 



7180 
7701 



8743 



9264 
9785 



0827 
1348 



1869 
2389 
2910 

343 » 
395 » 



4472 



1043 
1566 
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2612 

3»35 



3658 
4181 

4703 



1095 
1618 
2141 
2664 
3187 



37"^ 
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5226 5279 
57495801 



6272 



6794 6846 6899 695 1 



7317 

7839 
8361 



88S4 8936 8988 
940694589510 
992899800033 
045005020555 
1024 



0972 



H94 
2016 

2538 
3060 

3582 



4103 
4625 
5146 
5668 
6189 



6711 

7232 

77Si 



8796 



4756 4808 4860 



6324 



73697421 



7891 



841484668518 



4155 
4677 



6241 



6763 
7284 
7805 



82748327 



49404992 



6449 6501 



T1 



1 



5512 



5980 6032 



6553 



93«7 
9838 

0358 
0879 

1400 



93699421 
9890 991.2 
04110463 



1921 

2442 
2962 

3483 
4003 



6085 
6605 



8848 



0931 
1452 



"47 

1670 

2193 
2717 

3239 



8 



120011252 
1723 1^75 



3762 
4285 



533« 
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63766428 



7943 



1077 



1546 

20682121 
2590 2642 

3112 



3164 
3634 3686 



4208 
4729 



51995251 

57205772 



6815 
7336 
7857 



2246 
2769 
3292 



3815 
4338 



5383 
5906 



7473 
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9040 9093 



9563 



1599I1651 
2173 
2695 



3216 
3738 



4260 
4781 

5303 
5824 



6294 6346 6398 



83798431 



8900 



0983 
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»973 

2494 
3014 

3535 
4«>55 



4524 , 
5044 5096 

55645616 



V 



1 



6137 
6657 



2025 

2546 

3066 

3587 

4107 



45764628 



7910 



8952 



2298 
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3344 



3867 
4390 
49^3 
5435 
5958 



6481 
7003 
7526 
8048 
857d 



9615 



00850137 
0607 0659 
1129 1181 



1703 
2225 

2747 
3269 

3790 



4312 

4833 
5355 
587^ 



68676919 
73887440 



7962 

8483 
9004 



9473 

9994 
0515 

1036 

1556 



2077 
2598 
3118 

3639 
4«59 



4680 



9525 
0046 

0567 

1088 

1608 



2129 
2650 
3170 
3691 
42 1 1 



D 



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4732 



514852005252 



5668 



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5720 



6189 6241 
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61 
62 
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72 

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241 

75 
76 

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78 
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9268567 
9081 

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8618 
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9270109 0160 02 II 



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1136 
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3704 



42174268 



4730 
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6270 
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8470 88348885 



9347 
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9280372 

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1909 

2422 



3446 



3959 
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5495 

6007 



6518 
7030 

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8054 

8565 



90779128 
95889640 
92901000151 
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1123 



1634 
2145 
2656 
3167 
3678 



0674 



0725 



1 187 
1701 
2215 
2728 
3242 



3755 



4782 

5295 
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68346885 



7347 
7860 

8373 



9398 
9911 
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1448 
1961 

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2934 2985 



3498 



4010 
4522 

5034 
5546 
6058 



6570 
7081 

7593 

8105 

8616 



066207 
1174 



1685 
2196 
2707 
3218 

37*9 



8721 



8670 

91849235 

9698 



1239I 

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2266 

2780 
3293 



3806 



3909 

432o|437 114422 

48844935 



4833 
5346 

5859 



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7398 
79H 

8424 



8937 

9449 
9962 

0475 



0936 0987 



1500 

20I2 

2524(2 

3037 

3549 



4061 



6621 
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1225 



1736 
2247 
2758 
3269 
3780 



3 

9749 
0263 

0777 



8773 
9287 

9800 

0314 
0828 



1290 
1804 
2317 
2831 

3344 



3858 



5397 
5910 



1342 

1855 

2369 
2882 

3396 



5449 
5962 



64246475 
6937 6988 



7449 
7962 

8475 



8988 9039 
955- 



9501 
0013 
0526 
1038 



2063 

576 
3088 

3600 



4112 



457346244675 
5085 5136 5187 

5597 5648 5^99 
610961606211 



7644 7696 
81568207 
866818719 



917992309282 

9691 974*19793 
0202 0253 



1787 
2298 
2810 

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7501 
8014 
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0065 

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1602 
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2627 

3»39 
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4163 



66726723 
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7747 
8258 

8770 



0304 
07650816 
1276 1327 



1838 

,^350 
2861 

3372 
3883 



93389389944* 



0366 
0879 



8824 



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1907 

2420 

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3447 



3960 



4474145*514576 

4987 
5500 

6013 



6526 
7039 

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8065 

8578 



9090 
9603 
0116 
0628 
1141 



1653 
2106 
2678 
3190 
3702 



6774 
7286 

7798 
8310 

8821 



1889 
2401 
2912 

34*3 
3934 



8875 



9903 
0417 

0931 



9955 
0468 

0982 



144414961547 



1958 
2471 
2985 

3498 



4012 



41H4 
4628 4679 

5038 5089 5 141 5192 

5551560356545705 

606^11611661676218 



6577 6629 
70907142 
7603 7655 
81168167 
8629 8680 



9142 



1705 
2217 

2729 

3*41 
3754 



4215 4266 

4727 4778 
52395290 
57515802 
6263631 



6826 

7337 
7849 

8361 

8872 



933393849435 
9844 9895 9946 
03560407 0458 

086709180969 
1378 1429 1480 



1941 
2452 
2963 

3474 



7 



8927 



8978 9030 

949* 9543 
00060057 

05201057 1 
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2009 

*5*3 
3036 

3550 



4963 



9193 



9654I9706 
0167 
0680 
1192 



0218 
0731 
1243 



1756 
2268 
2780 

3*93 
3805 



4317 



534» 

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46365 



6877 

7389 
7900 

8412 
89*3 



35*5 



3985 4036 



2o6i 

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3088 

3601 



^680 
7193 



6731 
7*441 



77067757 



8219 
8732 



9244 9296 

9757 9808 
0270 
0782 
1295 



1807 

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2832 



3856 



4368 



482948804931 



539* 
5904 

6416 



795 > 

8463 



1992 2043 
2503 25 
3014I3005 

3576 



4087 4 



wmm 



n 



1085 



1598 
2112 

2625 

3»39 
3652 



166 



8270 
8783 



0321 

0833 
1346 



1858 
2371 
*883 



3344 3395 



3907 



4419 



5443 
6467 



6928 6979 
74407491 



8003 
8514 



8975 9026 



94869537 
9998 00A9 
0509 0500 
1020I1071 
1532 



1585 



2094 



554 *6o5 
06c ^116 



3116 

3627 

138 



5^ 



I 

s 

3 

4 
S 
6 

7 
8 



5 
10 
16 

XI 

16 

5 
3^ 
4» 
47 



I 
1 

i 

4 

6 

7 
8 

9 



5 



s 

10 

so 
z« 

3> 

3« 

4" 

4<S 



DiFIT 



N'.'Sso ^'9^^ 



OF Numbers. 



N 



8500 
01 
02 

04J 



05 
06 

08 
09 



8510 
II 
I 

13 
»4J 



9296 
9806 
2I9300316 
0826 
1336 



18 
19 



8529 
21 
22 

*3 
24 



«7 
28 

29 



8S3o| 
3» 

32 
33 
341 



36 

37 
38 
39 



8540 
41 

43 
44 



45 
46I 

47 

48 

«4? 

N 



9294189 
4700 
5211 
5722 
6233 



6743 

7254 

7764 
8275 

^7^ 



1847 

^357 

2 

3376 
3886 



866 29 



4396 
4906 

54*5 
5925 
6434 



6944 

7453 

7963 
«472 

8981 



9490 954 



9999 
9310508 

1017 

1526 



ao35 

*544 
3053 

3562 

4070 



4579 
5087 

559<^ 
61a 

6612 



7121 
762 

«»37 
9«53 



4240 

475 » 
5262 

5773 



62846335 



6794 6845 



7305 

78.15 
8326 

8836 



9347 

9857 
0367 



0877 0928 



1387 



1898 
2408 

'7 
3427 
3937 



4447 

4957 
5466 

5976 

6485 



6995 

7504 
8014 

8523 

9032 



0050 

0559 
1068 

*577 



2086 

^595 
3104 

3612 

4121 



4630 
S138 
5647 



7171 
97680 



4291 
4802 

53*3 
5824I5875 



4343 4394] 
480248534905 



7356 
7866 

8377 
8887 



9398 
9908 
0418 



1438 



9449 

9959 

o 

0979 
1489 



1949 2000 
2459 25 



2968 

3478 
3988 



449845494600 



5008 

55»7 
6027 



5059^5 
5568 

6078 



7046 
7555 



9083 



1119 
1628 



2«37 
2646 

3663 
14*72 



4680473 



7222 

7731 
8239 

8747 



8ia8 
8696 

9204 925 5J9306 



5364 



6386 



6896 6947 



74^7 

79«7 
8428 

8938 



46905 



10 
5019 
3529 



4039 4x590 



65366587 



7097 



80648145 

85748625 

91 



349 



9592 9643 
Old 0152 
06100661 



1170 
1679 



2188 
2697 
3205 

37 » 
4223 



5189 
5697 
62066 

67146765 



5240 
5748 

257 



7273 
7781 
8289 
8798 



54»5 
5926 

6437 



7458 
7969 

8479 

8989 



9500 
0010 
20 
1030 
1540 



2051 
2561 

3070 
3580 



no 
5619 
6129 
6638 



7148 



76067657 



8166 

8676 

185 



9694 
0203 
0712 
1221 
1730 



2239 
2748 
3256 

43765 
4274I 



4782 
5291 

57991 

6307 

6816 



7324 
7832 

8340 

8848 

9356 



3 r 4 



4445 

4956 
5466 

5977 



6488 6539 6590 664 



6998 
750917 



7050 

» 

8o2o|8o7 X 



8530 
9040 



9551 
0061 

0571 

1081 

1591 



2102 
2612 
3121 
3631 
4141 



4iS5i 
5J16 
5670 
6180 



05 



7199 



8217 
8727 
9236 



9745 
02 

0763 

1272 

1781 



2290 

2798 

3307 
3816 



737S 
7883 

8391 

8899 

9407 



4496 
5007 

55»7 
6028 



4547 
5058 

5569 
6079 



6076 



8581 
9091 



9602 
0112 



06220673 



1132 
1643 



2153 
2663 
3172 
3682 
4192 



2204 

2713 
3223 

3733 
4243 



4702 

211 

5721 

6231 



^753 
5262 

5772 
6282 



6689 6740 679 



770877597810 
82688319 
8777 8828 



540305 



081 

1323 
1832 



2341 
2849 

3358 
3867 



43244375 



4884 4935 



4833 
534» 

58501 _., 

63586409646 
6867 ' 



5392 
5901 



6917 



7426 

7934 
8442 
8950 

9458 

61 



7101 
II 
8122 
8632 
9142 



9653 
0163 



1183 
1694 



7250 7300 



92879338 



9796 9847 



0356 



40865 



«374 
1883 



2391 
2900 

3409 
3918 
4426 



5443 
5952 

o 



7476 
7985 

8493 
9001 

9509 



4598 4649 



5109 
5620 
613c 



71527203 
76627713 
8173 8224 
8683 8734 
91949245 



97049755 
09140265 

07240775 

X2341285 

1745 1796 



2255 
2764 



32743325 



3784 
4294 



48044855 



53»3 

58^3 

6533 
6842 



2442 
2951 
3460 
3968 

4477 



4986 

5494 
6002 

65 1 1 

6968I70X9 



7527 
8035 



9052 
9560 

.8 



(IS7) 




5160 
5671 
6181 
6692 



2306 
2815 



3835 
4345 



5364 

5-874 

6383 
6893 



7402 
7912 
842.1 
8930 

9439 



^949 
0458 
0967 

H75 
1984 



2493 
3002 

3511 

4019 

4528 



5036 

5545 
6053 

6562 

707q 



7578 
8086 



8544 8594 



9102 
9610 



S» 



S» 



I 

3 

4 

5 

4 

T 
8 



5 

tc 

15 

10 
26 

3« 
3« 
41 
46 



50 



r 

% 

3 

4 

s 

7 
8 



5 

ic 

10 

3^ 
35 
4C 

45 






l (i6o) 



Logarithms 



TJ 



N. 865 L;'937 



8650937016110211 

51 ,06630713 

52 1x65 1215 

53 1667 1717 

54 116 92219 



I8660I 
61 
62I 

63 
64^ 



57 
59 



$5 

66 

67 

68 

69 



75 

76 

77 
78 
79 



5*79 

c68o 

6182 

6683 
7184 



0261 
0764 
126^ 
1767 
2269 



4671 
3172 

J674 3724 
417614226 

4677 



7686 7736 
8187 8237 
8688 8738 
91899239 
9690 9740 



2721 2771 

3«3 3273 

3775 
4276 

4728I4778 

5^79 
5781 

6282 

67S3 

72_85 

7786 
8287 
8788 
9289 
9790^ 



5229 

S73» 
6232 

6733 
7^39 



86709380191 

71 0692 

72 1193 

73 »693 

74 *i94 



2695 
3696 



4697 



0312 
0814 
1316 
1818 

23»9 



2821 

3323 
3825 

4326 

4828 



5329 
5831 

6333^ 



7335 



0241 
0742 
1243 

1744 
2244 



^745 
3245 
3746 



41964247 



4747 



R6Soi 

8x 
82 

«3 
841 



8690 9390198 0248 



H 

86 

«7 
S8 

89 



91 

93 
94 



95 
9& 

97 
98 

99 



5^97 
5698 



6x986248 



6698 
7198 



7698 
8198 
8698 



5^7 
5748 



0291 
0792 
1293 

*794| 
2294 



2795 
3296 

3796 

4297 
4797 



6748 
7248 



5297 

5798 
6298 

6798 

7298 



7748 
8248 

8748 



91989248 



969 



8 



0697 
1197 
1697 
2196 



-22 

Nl 



2696 

3»95 
3695 

4194 
4^3 



9748 



P747 
1247 

»747 
2246 



2746 

3HS 

3745 
4244 

4743 



0362 
0864 
1366 
1868 

237c 



2871 

3373 
3875 
4376 
4878 



5380 
5881 
6382 



68346884 



7385 



78367886 



8337 
8838 



93399389 
9840 9890 



034» 
0842 

«343 
1844 

^344 



2845 
3346 



4347 
4847 



5347 
5848 
6348 

6848 
7348 



8387 
8888 



0391 
0892 

«393 
1894 

12394 



2895 
3396 



38463896 



14397 
4897 



0412 
0914 
1416 
1918 
2420 



2922 

3423 

392s 

4427 
4928 



543^ 

5931 
6432 

6934 

7435 

7936 

8437 
8939 



0441 



7798 
829S 

8798 



9298 9348 



9798 



0298 0348 



P797 
1297 

"^797 
zzgS 



2796 

3295 

379S 
4294 

4793 



7848 
8348 
8848 



9848 



0847 

«347 
1847 

2346 



2846 

3345 
3845 



5397 
5898 

6398 

6898 

7398 



7898 
8398 
8898 

9398 
9898 



0398 
0897 

1397 
1897 

2396 



2896 

3395 
3894 



4344 4394 
4843I4B93 



1 



0462 05 1 3 0563 061 3 



0964 
1466 
1968 

2470 



2972 

3474 
3975 
4477 
4978 



5981 

6483 
6984 

7485 



7986 
8488 
8989 
944019490 



9941 



09420992 



>443 
«944 
2445 



2945 
3446 
3946 
4447 
4947 



5447 
5948 
6448 

6948 
7448 



9991 



T 



1015 
1516 
2018 
2^20 



3o»« 



35»4 3574 



4025 
4527 



5028 5079 



54805530 



6031 

6533 
7034 

7535 



8037 
8538 

9039 

9540 
0041 



0492 0542 



1493 
1994 

2495 



2995 
3496 

3996 

4497 
4997 



S497 
5998 

6498 

6998 

7498 



7948 

8448 

8948 
9448 
9948 



1065 
1567 
2069 
2570 



3072 



4075 
4577 



6583 
7084 

7585 



9 \i^ 



1617 
2119 
2621 



3122 

3624 
4126 

4627 

5i£9 
5580I5630 
60826132 



6633 

7*34 
7636 



8087 
8588 
9089 
9590 
0091 



8137 
8638 

9»39 
9640 

014X 



1042 

>543 
2044 

2545 



0592 
1093 

>593 
2094 

2595 



3045 

3546 
4046 

4547 
5047 



5547 
6048 



6548 6598 6648 



7048 
7548 



0448 



0947 0997 



1447 

1947 
2446 



2946 

3445 
3944 



7998 
8498 
8998 
9498 
9998 



0498 



H97 

1997 
2496 



2996 

3495 
3994 



4444 4494 
4943 4993 



8048 
8548 
9048 



0548 
1047 

1547 
2046 

2546 



3045 
3545 
4044 



5043 



0642 

"43 

1643 

2144 
2645 



3095 
3596 
4096 



5097 



5598 
6098 



7098 



8098 
8598 



9548 9598 9648 



0048 0098 



Fro 



5» 



I 

1 

3 
4 
s 

16 

7 
1^ 



5 

ic 

15 

20 
x6 
3« 

4« 
46 



3145 
3646 

4146 



4597 4647 



5H7 



5648 
6148 



7148 



75987648 



8148 
8648 



90989148 



0x48 



0598 0648 



1097 

»597 
2096 

2596 



3095 
3595 



4544 4593 



5093 



» 



"47 
1647 

2146 

2646 



3H5 
3645 



40944144 



4643 
5»43 



^ 



50 



I 

tx 

3 
4 

5 

6 

7 
8 



50 
5 
ic 

15 
so 

»5 

30 
35 

40 
4J 



D 



Pts 



N> 870 L, 939 

IT 



3700 
01 
02 
03 



06 

07 
08 



87109400182 



II 

12 

>4 



■7 
18 

»9 



4720 
21 

' 22I 

«3 
241 



26 

*7 
28 

*9 



36 

37 
38 
39 



874<^ 
4« 
42 
43 
44 

46 
47 
48 



939S>93 
5692 

6191 

6690 

718$ 



7688 
81S7 
8685 
9184 
9683 



0680 07 Jo 0780 0830 



1179 
1677 
2176 



2674 
317^ 



3670 3720 



4169 
4667 



5165 
6161 



665967096758 



7157 



7654 
8152 

8650 

9H7 

9645 



87309410142 
31 0640 

33 «635 

34 ^2132 



2629 
3126 
3623 
4120 
4617 



SIM 

561 1 



6605 
7101 



7598 
8095 

8591 

9088 



J2i54 9^ 



5242 
5742 

6241 

6740 
7239 



7738 
^237 
8735 

9234 

9733 



02^1 



1229 
1727 
2225 



272^ 
3222 



4218 
47«7 



5215 

57»3 
6211 



7206 



7704 
8202 

8700 

9197 

9695 



0192 
0690 
1187 
1684 
2182 



2679 
3176 

3673 



4667 



5661 



6108 6158 



7648 
8144 
8641 

9»37 



5292 

5792 
6291 

6790 

7289 



7788 
8286 
8785 
9284 

9783 



0281 



1278 
2275 



2774 
3272 

377^ 
4268 



47664816 



52645314 



5762 



626063106360 



7256 



77541 
82^2 

8749 

9247 



0242 
0739 
1237 

173* 
2231 



2729 
3226 

3723 



41704220 



47^7 



51645214 



57" 
6207 



6654 6704 6754 



7151 7201 



7697 

819^ 
8691 



OF Numbers. 



5342 

5841 

634J 

6840 
7339 



7837 
8336 

8835 

9334 
9833 



0331 



1328 
1827 

2325 



2823 

3322 
3820 
4318 



5812 



6808 
7306 



8301 
8799 
9297 



9744 9794 9844 



0292 
0789 
1286 
1784 
2281 



2778 

3275 
3772 



4766 



5263 
5760 
6257 



7250 



7747 
8244 

8740 



5392 
891 

6390 

6889 

7388 



7887 
8386 
8885 
9384 
9882 



0381 
0880 
1378 
1877 

2375 



2873 

3372 
3870 

4368 

4866 



5364 
5862 



6858 
7356 



78047853 



8351 
8849 
9346 



0341 
0839 

1336 
1834 

2331 



2828 

3325 
3822 



42704319 



4816 



53>3 
5810 

6307 

6803 

7300 



91879237 
968JI9733 

a 3 



7797 
8293 
8790 

9286 
9783 



5442 

594« 

6440 



6939 6989 



7438 



7937 
8436 

8935 



9434 9483 



9932 



0431 
0929 
1428 
1926 
2425 



2923 
3421 
3920 
441.8 
4916 



54H 
5912 

6410 

6908 

7405 



7903 
8401 
8899 



0391 



1386 
1883 
2380 



2878 

3375 
3872 

4369 

4866 



5363 
5860 



6853 

7352 
7846 

8343 
8840 



5492 

599» 
6490 



7488 



7987 
8486 

8985 



9982 



0481 
0979 
1478 
1976 

2475 



2973 
347 > 

3969 
4468 

4966 



5464 

6460 
6957 

7455 



7953 
8451 

8948 

9396 9446 9496 
9894 9943 9993 



0441 



08890938 



1436 

»933 
2430 



2927 

3425 
3922 

44«9 
4916 



5412 
5909 



63566406 



6903 
7399 



5542 
6041 



6540 6590 



7039 
7538 



8037 
8536 

9035 

9533 
0032 



0531 
1029 
1528 
2026 

2524 



3023 
3521 
4019 

45»7 
5015 



55*3 
601 1 



8003 



8998 



0491 
0988 
1485 
1983 
2480 



2977 

3474^ 

397 » 
4468 

4965 



5462 

5959 
6456 

6952 

7449 



"•* 



7896 

8393 
8889 

93369386 



9832 



T 



5592 
6091 



7089 
7588 



8087 
8586 
9084 

9583 
0082 



0580 

1079 

»577 
2076 

2574 



3073 

357> 
4069 

4567 
5065 



6061 



650965596609 
7007 7057 7107 
750575557605 



8053 



8500 8550 8600 



9048 

9545 
0043 



1038 

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2032 

2530 



3027 

35M 
4021 

4518 

5015 



5512 
6009 
6505 
7002 



8442 

8939 

9435 
9882 9932 



7946I7995 
8492 



5642 
6141 

6640 

7^39 
7638^ 



8137 
8636 

9^34 

9633 
0132 



(161 ) 



0630 
1 1 29 
1627 
2126 
2624 



3122 

3621 

4119 
4617 
5115 



5613 

6III 



8102 



9098 

9595 
0093 



0540 0590 



1088 
1585 
2082 

2579 



3077 

3574 
4071 

4568 

5065 



5562 
6058 

6555 
7052 



7499 7548 



804s 

8542 

8988I9038 



9485 



8 



9535 



9981 0031 



JL 



so 



7 

9 



S 

10 

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40 
45 



49 



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5 

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5 
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D 



Pu 






fR" 



5» 

53 
5^ 



f -^1 



I8750I94200S110130 01 80 a2M 0279 
0577J0626 0676 0726 0775 



1073 1123 



55 
56 
57 
5« 



1569I 

2065 



2562 
3058 

3553 
4049 



I 



1172 

t6i9|i669 
2115I216; 



8760 
61 
62 

63 
64I 

65 
66 

67 
68 

69 



2611 

3107 
3603 

454S 4595 



8770 

71 
7* 
73 
74 



2661 

3>57 
3653 
4149 



5 



1222 1272 
1718 1768 
2214 2264 



5041 5091 

5537 ?5»6 
6032I6082 



6j28 6578 6627 



70247073 



7519J7569I7618 
8015 



8510 
9005 
9501 



76 

77 
78 
79 



9430491 
0986 
1^8 
1976 



4644 4694 4744 



5636 
6132 



27102760 
3206 3256 
37023752 
4198 4248 



0329 
0825 
1321 
1817 
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H 



5 140 5 190 5239 



5686 
6181 



7123 



80648114 
8560 8609 



905 s 



9996 0045 



95509600 



7668 
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5735 
6231 



037^ 
0875 

»37J 
1867 

2363 



2810 
3306 
3801 

4297 
4793 



f28 0478J0 

J24 0974 1 



042810478105271 

0924 0974 1023 
1420 1470 1520 
1917 19662016 

2413246212512 



2859 2909 



6677 6726 

7172 



7222 



7717 
8213 



8659{87o8 



910491549204 



9649 9699 



0095 



0144 



P54I 
1036 

1531 

2026 



0590 0640 



8780 
81 
82 

83 
JB4J 

^1 
86 

87 
88 

89 



2471 
2966 
3461 

3956 



4945 



2521 
3016 
3510 
4005 



1085 
1580 
2075 



2570 
3065 
3560 



1135 
1630 
2125 



3115 
3609 



6776 
7271 



01941 
0689 
1184 
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20 



041 39,2685 1 ,58225,0407 J 
0791 8, 1 2460,4762^,82772 
1 1394,33523,06836,76921 
14612,80356,78238,02593 
17609,1 2090,55681, 2420^ 



21 
22 

*3 

2 

"26 

*7 
28 

*9 



< 



3' 

3» 
33 
34 

ii 



m* 



Logarithms 



Mi*Wi*i*i«* 



■**i* 



2041 1,99826,55924,78085 
23044,89213,78273,9281:4 
25527,3505 1,03306,0698c 
27875,36009,52828,96154 
30102.99956,63981,195^ 



32221,92947133919,26801 
3^242,26808,22206,23596 

36172,78360,17592,87887' 
38021,12417,1 1006,02294 

^^l39794>ooo^o»7^037*^<^S7 



4H97»33479»70^ » 7»9644« 

43 136,37641,58987.3 « "5 
44715,80315,42219,22114 

46239,79978,98956,08733 
12,12547,19662,43730 



30477 



4913^16938,34272,67967 
50514,99783,19905,97607 

p85«>39398-77887,47»05 

S3«47.89i70*42255»»«375 
S44oo>go443,50275,635iO 

3^ 5£630,25oo7,6^287,26j02" 

38^ 

39 
.0 



56820, 1 7240,66994,9968 1 
57978,35966,16810,15675 
59106,46070,26499,20650 
0205,99913,27962,39043 



41 61278,58567,19735,49451 

42 62324,92903,97900,46322 

43 63346,«f 5J5»79586,5264i 
44 



^434$ 



,26764, 



86187,43118 



4S6»2l,2<l37,7S343>67938 



4666 



^75»7^3>Mi574»074o8 

47 $7209>7^579.3S7»7»4^44» 
i^ 68«4,i2373,75587,2i8i5 

49 69M9;6o8oo,285 1 3,66442 

169897,00043,36018,80479 



ill 

J7 



1^ 

6"i 
62 
63 
64 
65 



N Logarithms 



5 M707 5 7 »o 1760,97936736584 
J^ 71600,33^36,34799,15963 

53 72427,58696,00789,04563 

54 73239»37598.i2968,5O7i0 



748 1 8,80270,06200,41635 

-.75587.4S5S6.7J»49«»39883 
5» 76342»79935>62937,2825 J 

77085,201 16,^2144,19026 

77815,12503,83643,63251 



66 81954,39355,41868,67326 
6^ 82607,48027,00826,4341 5 
68 83250,89127,06^36,31897 
6g 83884,90907,37255,31616 
701 84509,80400,14256,8307 



78 

79 
80 



97 
98^9 

99 
100 



74036.26894,94243,8455^ 



78532,98350,10767,03389 
79239,16894,98253,87488 

79934*05494' J358^7°530 
80617,99739,83887,17128 
81291,33566,42855,57399 



71 85125,83487,19075,28609 
7* 857'33,24964,3 1268,46023 
73 86332,28601,20455,90107 
74^6923t»7J97»30976>»92oi 
75 87506,126/3,91700,0468;^ 

76 



88081,35922,8079143519 

771^8649,07251,72481,8714(3 

89209,46026,90480,40172 

89762,7091.2,90441,42799 

903o8,99869f9'943»S85^^ 



81 90848,50188,78649,74918 

82 91381,38523,83716,68972 

83 91907,80923,76073,90383 

84 92427192860,61881,65843 
85 192941,89257,14292,73333 



36|93449»845***43567»7^»6* 

87 93951,92526,18018,52^63 

88 94448,26721,50168,62039 
94939,00066,44912,78472 
95424,25094,39324.87459 



89 
90 

91 
9« 
93 
94 

IS 
96I98 



95904,15923,21093,59992 
96378,78273,45555,26930 
96848,29485,53935,1 1696 
97312,78535,99698,65963 
9777g>36o52,88847,76632 



227,12330,39568,41336 

98677,17342,66244,85178 

122,60756,92494,85664 

99563-5»945»97549.9J534 
00000,00000,00000,00000 



A a a 



I(IM) 



4^ 



Loo AXimut 



mm 



Tab. II 



N 



i 



oi 90432.13737,8^642,57428 

02 oo86o,oi7i7«6i9i7,56i05 

03 01283,72247,05172,20517 

04 01703,33392,98780,35485 
021 1 8.92990^69938,07279 



Logarithms 



?5 

06 02530,58652,64770,24085 

07 02938,37776,85209,64083 

08 03342,37554,86949,70231 

09 03742,64979,40623,63520 

10 04139,26851,58225,04075 



" 04532,29787,86657,43410 
1204921,80226,70181,61157 

«3 o5307»84434.834«9»7"«o 

14 05690,485*3.36472,59405 
15106069,78403,53611,68365 



16 06445,79892,26918,47776 

17 06818,58617,46161,64380 
1807188,20073,06125,38547 

'9 o7S54»696i 3,9*530.75925 
20 07918,12460,47624,82772 



21 

22 



25 

l6 

?7 
28 

*9 

30 



08278,53703,16450,08150 
08635,98306,74748,22910 

23 08990,51114,39397,93180 

24 09342,16851,62235,07009 
09691 ,001 30,08056,41436 



•^^m 



10037,05451,17562,90052 
10380,37200,55956,86425 
107^,99690,47808,36650 
1 1058,9710^,99248,96370 
»>394»335*3»p6836,7692i 



31 1 1727,12956,55764,26081 

32 1205:7,39312,05849,86847 
33 12385,16409,67085,79225 
34|i 27 1 0,47983,64807,62936 

35 

36 

37 

38 

39 

4? 
4» 
42 
43 



' 3033*37684,95006,1^667 

i3353»89o83,702i7,5i4i8 
1 367 2,0567 1 ,56406,76856 
13987,90864,01236,51138 
14301,48002,54095,08046 
4612,80356,78238,02593 



45 
46 
47 
48 
49 



14921,91126,55379,90171 
15228,83443,83056,48131 
« 5 5 3 3,60374,65061 ,80996 
1 5836,24920,95249,65545 
10136,80022,34974,89212 

164315,28557,84437,09649 
i673i,73347,48i76,09872 
17026,17153,94957,38724 
17318,62684,12274,03826 
J 7609^ 1 2 <; 90,^^68 1,24208 



N 



5« 17897,69472,93169^3687 
5* 18184,35879,44772,54718 

53 18469,14308,17598,80313 

54 18752,07208,36463,06668 



55 
56( 



57 
58 



Logaritluiis 



19033,16981,70291,48445 



I93«.45983»5446i,59693 
19589,96524,09233,73676 

1 9865 , 70869, 54423,6232 1 

591201 39,7 1 243,2045 1 ^48293 

20411.99826.55924^78085 

61 
62 

63 

64 

^5 

661 

67 



20682,58760,31849,70958 
2095 1 ,50145,42630,94439 
21218,76044,03957,80764 
21484,38480,47697,88494 
2 1748.39442, 1 3906,28283 



22010,80880,40055,09905 
22271,64711,47583,27998 
0,92817,25862,85365 
22788,67046,13673,53841 
23044,89213,78273.92854 



682253 

69 

70 



7' »3*99.6i 103,92153,83613 

72 23552,84469,07548,9i«3 

73 23804,61031,28795,41456 

74 H054.9248a»82599,7i984 

75 24303,80486,86294,44028 



7^ 2455 



1,26678,14149,82161 

H797»3a663,6i8o6,62756 

,00023,08893,97994 

,30309.79893,16957 

25527,25051,03306,06980 



77 
7825042 

7925285 

80 



81 

82 

83 
8^ 



25767*85748,69184,51029 

26007, 1 3879,85074,795 ' 3 
26245,10897,30429,471 18 
2648 1 ,78230,095 36.4645 1 
26717,17284,03013,80159 



86 26951,294^2,17916,31218 

87 27184,16065,36198,96929 

88 27415,78492,63679,85484 

89 27646,18041,73244,14260 

90 27875,36009,52828,96154 

91 28103,33672,47727,53764 

92 28330,12287,03549,60858 

93 28555,73090,07771,76060 
94I28780, 1 7299,10226,04700 

95 



96 



97 
98 

99 



29003,461 13,62518,01 129 



29225,60713,56476,051 



g 



29416,62261,61 592,92737 

29606,51902,61531,11055 

29885,3076^09706^65010 

00^30102,99956,63981.19521 



its: 



N 



II. 



to aO rLAC£8. 



2pi 

202j 
203 
204 
205 



30319,60574,20488,87144 
30535,13694,46623,76949 
307^9,60379,13212,91805 
30963,01674,25898,75626 

206 31386,72203,69153^40038 

207 3«597»o3454.569«7.75346 

208 31806,33349,62761,55006 

209 32014,62861,11054,00229 

210 3ag»«»9»947>339iQ>268oi 

211 3 2428,245 5 2,97692,66508 

212 32»33»5S6o9«2S75if436o6 

213 3a837»96o34»38737»7^339 

214 33<^4>.37733*49»9<'*836o5 

«i5 33243.84599»»56o5,33iJ9 



216 
217 
218 
219 

220 



221 
222 
223 
224 

22 

226 

227 
228 
229 
230 



?4» 
242 

H3 

*44 



z^6 

H7 

248 

«49|3 
250 



LfOgtnthms 



33445'37S"»S093o»89753 

33045»97338.48529»S»o38 
33845«^4936,046o4,8304i 

34044,41148,40118,33837 

34242^26808,22206,2 1 ^^ 06 



225352 



34439,22736,85110,69775 
34^35»«9744»5<5638,62932 
34830,48630,48 160,67348 
35024,801